Please post the tractor model number and the serial number from the data plate or decal on the chassis.
This will allow us to go online to view an owners manual.
It's possible that the bare wires you have found made contact with a grounded metal surface and is killing the spark from the magnetos.
You need to tape up the bare wires, or better yet, replace the damaged sections with new wire.
The engine may have suffered some real damage from overheating caused by the mouse nests.
But we can walk through the basics before exploring that, and right now.....basics would be to post the model and serial numbers........and tape up all the bare places on the wires you found.

The model # is 13BX11CG709, and the serial # is 1K234H30154

Click link and download your free copy of the service manual.
It's too late for me to do anything else (except get to bed).

Here is a link that might be useful: SV590S service manual

Got it. Thanks!

Because you have found proof of rodent activity, the pests may have damaged other wires, or the same wires in other places, so you need to trace the wires as far as you can to determine if there are any more bare places that could be grounding the wires. Grounding the wires you discovered is how the engine is stopped or killed when you turn off the key switch, or when you arise from the seat (under certain conditions).
To absolutely rule out whether the wires are grounding and preventing the ignition system from making spark you can do the following test. You will need to beware that if the engine starts and runs during this test, you will need to pull the choke, or disconnect the wire to the fuel solenoid (on the carb) in order to kill the engine. So, keep that in mind before you do the test because you DO NOT want to run the engine more than about a minute or so with the blower housing removed.
Trace the wires to each magneto. Carefully remove the wire from the terminal on each magneto and place the wire away from the flywheel. Try to start the engine using your normal procedure.
If the engine starts and runs now, there is a problem somewhere in the kill wires circuit, or maybe in the safety interlock system circuits.

If the engine still remains in the non-running state, give the carb a shot of carb cleaner (as you did before) to rule out lack of fuel as a second (unlikely) cause of non-run.

If the engine still does not run after this, remove the spark plugs and hook the plug cable back to the plug. Lay the plug on a metal part of the engine (not the flywheel) where you can observe the spark gap.
Crank the engine and watch for spark. If you see a good spark (you might even hear it snap), alas, the engine has probably sustained some serious damage from the overheating.

As you mentioned, this engine DOES have ACR, so a standard compression test is not recommended for determining the condition of the cylinder combustion chambers.

A cylinder leakage test (described in the Kohler manual), using compressed air can pinpoint where the loss of compression is occurring. But a cylinder leakage test requires that the individual cylinders be positioned at a specific spot in the 4 stroke cycle or the test results will be misleading or downright useless.

Complete the tests of the ignition system and post back your results.

The mice pretty much filled up the entire space under the plastic cover with fluffy nesting stuff. At first I wasn't sure if the fluffy stuff was supposed to be there, since there was so much of it. But now I'm pretty sure it's not supposed to be there, and it's interfering with engine cooling. I looked at the wires again, more closely this time. And it does look like they've been chewed. But it looks like that happened a long time ago, since someone put electrical tape around it before. But that tape has fallen off. This tractor came with the house I'm renting, so I don't know much of its history. This is my 2nd year using it.

So, I disconnected the kill wire for the magneto thing--that wire was connected to the chewed stuff--and it didn't make any difference. Then I reconnected the kill wire, and squirted some carb cleaner in the carb, and tried again. After it turned over a bunch of times, I got one really loud pop. After my ears stopped ringing, I took out the spark plug. Put my thumb over the hole, and turned the flywheel by hand in both directions. I didn't feel much, if any, air pressure. But I heard something that sounded like air being released. That may have been normal operation of the ACR though. I didn't try checking the spark again, since I was a little concerned after the explosion from the carburetor. I figured I'd wait a while, and let all that stuff dissipate first. But I'm pretty sure, since I checked it before, that there's enough spark to cause the engine to fire at least a little. And it could be a normal spark as far as I can tell.

The next thing I checked was the fuel solenoid, since one of the chewed wires was attached to it. It clicked when I turned the key, so I think that's ok. The spark plug is dry, so I'm wondering if it's getting fuel. The fuel filter is only half full of gas, but I'm guessing that could be normal?

Here is what you have proved so far:
Fuel system.
The non-running state of the engine IS NOT because "the engine is not getting fuel". The shots of carburetor cleaner you administered IS FUEL, and if the non-run was due to a problem in the fuel system of the engine, the engine would have responded to the presence of the carb cleaner "fuel" by eagerly running........for about 1 or 2 seconds for each shot of carb cleaner.
Since the engine remains unresponsive even when given an alternate fuel source (carb cleaner shots), lack of fuel can be discounted. Cross fuel off the list.

Ignition system.
You have stated a couple of times that you believe the ignition system is producing spark. The bang that nearly deafened you (soiled BVDs?) proves that you at least had one instance of spark. So, we might want to presume (for the moment) that the ignition system is OK.
Disconnecting the kill wire from the magneto eliminated the possibility that chewed or bare wires were stopping the production of spark. Place a question mark or 2 beside "Ignition system" on your list.

You have indicated that you feel little or no compression when you place a thumb over the spark plug hole and turn the engine by hand.
Do this. (you might need a helper) Move the spark plug wire as far away from the spark plug hole and place your thumb over the spark plug hole. Spin the engine using the starter.
There should be enough compression that it takes a bit of effort to keep your thumb held over the hole. If you do not feel the compression trying to push your thumb off the hole.........the engine is not making or holding enough compression to run.

If this is what you find, it is time to decide if you want to go further, or whether you would even be able to go further depending on how well equipped you are to perform a cylinder leakage test.
Read and study the Kohler service manual page 3.4 about the cylinder leakdown test.
Though the Kohler leakdown tester kit is nice, you really don't need that much capability.
But you will need a source of compressed air, an adapter fitting for the spark plug hole to mate the compressed air hose to the engine cylinder head, and you must have socket wrenches large enough to fit the flywheel retention nut so the engine cannot move when the compressed air is applied to the engine.
If you qualify in regard to the above, and feel confident/mechanically competent enough to do a cylinder leakage test, then you can do one to see why the engine has insufficient compression.
E-mail me for a copy of my description of how to do a "poor boy's leakdown test". The poor boy's test still requires the above tools and compressed air, so if you do not have (or can borrow) those, you do not qualify for a DIY leakdown test. Place a "frowny face" next to "Compression" on your list and choose a professional small engine shop for further testing.

mownie - I already tried putting my thumb over the spark plug hole, and cranking the engine. If there was any pressure on my thumb, it was barely noticeable. I wasn't sure if that was from normal ACR operation. Or if something was leaking. Now I'm thinking something's probably leaking. I've been looking for a cheap push mower to get me through the rest of the season. And I may continue to work on the tractor later. But for now, I'm assuming it's a goner. Thanks a lot for your help.

It seems like the mice have won this time around.
Even with ACR you should feel a good deal of pressure on your thumb if the engine is healthy.
ACR only reduces compression, it does not eliminate compression.

Unfortunately nobody won this time. My tractor's dead. And so are the mice:( Too bad these things don't have screens to keep them out. At least I'll know better next time.

I just bought an old mower from craigslist, and I think it'll get me through the rest of the season. And I needed one anyway, so I could collect grass and weed clippings for my garden. I read through the tractor manual, and it sounds like the problem could be something like a stuck valve. So I'll probably try some kind of oil additive stuff, that might be able to unstick things. I figure it's worth a try.

I was using my Cub 1045 most of the day about 2 weeks ago and after parking it for 20 minutes I found it wouldn't start. This is the wrong time of year to be taking a mower to any repair shop so I set out to fix it myself. It would do "nothing" when I turned the key. No juice to anything. I used several suggestions on You Tube and replaced the battery, ordered and replaced the PTO, and was in the process of looking for a new ignition module when I ran across a post by a gentleman who professed to have "no automotive knowledge whatsoever" who advised his Cub had a fuse block just to the side of battery and he was "up and running" after inserting a replacement 20 amp fuse. I returned to my Cub and under the dust I too found a yellow 20 amp fuse. After a quick trip to True Value I was "up and running" also. I hope this was useful to you.

G Lawrence - a fuse blowing like that is just a "symptom" of a deeper problem. A fuse does not blow unless there is a reason - that being an excessive amount of current flow usually caused by a circuit short to ground.

Suggest you closely check your machine for any evidence of frayed/worn wiring and don't be too surprised if you do blow another fuse in the near future....

Mine wouldn't start after sitting for 2 months. Fired when shot with carb cleaner. Gas pumped out beyond fuel pump. Only thing left was to beat on the fuel bowl of the carb. After a couple of good smacks it started! Must have been a stuck float.