I agree with your diagnosis.

You can't make matters much worse! If you do decide to take apart the carb do it with a large table and lots of light. Use carb cleaner as solvent. Use toothpicks to clean small openings.

If this does not work look into finding a used working carb on ebay.

Remove the bottom of the bowl on the carburetor. Check if it has gas in it, if not your float seal is stuck closed. The float seal is the little square piece attached to the float. As the float drops the seal will open filling the bowl with gas. It happens when the gas evaperates out and the rubber tip sticks in the hole.

Wheelhorse and blacksmithman are both right. To me those carbs are real simple but I'm a mechanic by trade. If you have good mechanic understanding I would think you could do it yourself without trouble. Well lighted area to work in and then I'll put some of those blue paper shop towels on the bench and tear the carb down there. You can see those little parts against that blue towel and the towels are disposalable.

And if you have a digital camera...........take plenty of photos as you disassemble. Take photos from different angles BEFORE removing each part. That way you will be able to refer to the photos when you go through reassembly. I take photos even if I have a shop manual because I have found on several occasions that line drawings or photos in the manual don't explicitly depict certain aspects of how the parts are oriented.

Thanks for the tips, guys. I'm getting the impression that there are many parts involved and this may be more complicated than I anticipated. I have removed the carb from a Yamaha 125 dirt bike and replaced the main and pilot jets. But that seemed to be about all there was to it. It was not complicated at all. This carb sounds way more sophisticated.
I guess I've got nothing to lose be trying it. If I mess it up, or it does not fix the problem, it will have to go to the dealer. If I don't try it, it will have to go to the dealer. So I'll at least take the carb off and see what I think at that point.
Thanks again.

PS I'll let you all know whether I take it apart or not.

Well, I took the top half of the carb off today and cleaned it up. The tiny pinholes in what I assume was a jet, looked OK. I could see light through them. I reassembled and still won't start. Next I'll take the whole carb off, but I fear that it still won't start.
To test the solenoid, the online Kohler manual says to use a 12V power source. Is my battery trickle charger considered a 12V source? I'm a real boob with electricle stuff. I'm thinking maybe solenoid.

If the starter works the solenoid is good. A solenoid is nothing more than an electrically activeted on and off switch. When you aply 12 volts to the small terminal it connects the two big terminals which puts 12 volts to the starter and you said it spins over. Is it spinning over fast enough? Is the battery up good? Was the fuel bowel in the carb full of gas when you took the top off? The rubber tipped needle will stick in the seat and not let fuel into the carb, happens a lot when one has set up awhile. Kinda sounds like whats going on. You did make sure there wasn't water in the fuel bowel didn't you?

Thanks for the suggestions. The starter is spinning as fast as it always did. There was fuel in the bowl and there was no water.
The solenoid I'm refering to is the fuel shut-off solenoid, according to the online manuel. They say to take it off of the carb and apply a 12V source to test if the pin retracts in the solenoid. Should I do this? And of so, can my trickle charger be my 12V source for testing?

Yes, a 12V "trickle" charger is sufficient. Do not test in an area where gasoline vapors or fumes are present as you will create a tiny electrical spark. To "bench test" the fuel solenoid you attach the negative cable (BLACK) of the charger to the clean, unpainted threads of the solenoid body, then with the charger plugged up, touch the positive cable (RED) to the wire lead of the solenoid. If the solenoid is working properly, you will hear/feel a small "click" when the wire lead makes contact/breaks contact with the charger positive cable. You should also see the brass plunger tip of the solenoid "pull into" the solenoid body slightly when the wire is connected, and extend or "push out" when the wire is disconnected. I have seen these solenoids stick because the fuel bowl content evaporated in long term storage and the heavy varnish and gum keep the plunger from moving. The plunger moves out under spring pressure of a very small coil spring. The electromagnetic force of the solenoid has to overcome the spring tension to "open" the plunger when the ignition key is turned on to allow fuel to enter the carb main jet. (the part of the carb that the plunger touches is the main jet or a seat ahead of the main jet). If the plunger does not pull in when attached to a 12V power source, fill a soft drink bottle cap with spray carb cleaner and position the solenoid so that the entire plunger end is submerged in the carb cleaner. Allow the plunger to "soak" in the cleaner for about 15 minutes. After removing the solenoid from the cleaner, push the plunger in manually by pressing it against a piece of wood. If it pushes back out instantly, it is free now. If it stays pushed in, it is still gummed up. Put the solenoid plunger end back into the capful of carb cleaner to soak for 15 more minutes. After you get the plunger freed up so that it comes back out instantly, hook up to charger and try the electrical test again. It might be necessary to prompt the plunger to move by pushing in slightly with a finger tip. Once the pluger pulls in, it should stay in until you disconnect the positive cable, at which point it should "pop" back out.

I did mention a couple of posts up that I was a boob with electrical stuff, didn't I? A friend suggested using the mower battery as a source. Duh!!! So anyway, I used the battery and the solenoid tested out fine. The pin went back and forth as electric was applied.
I have cleaned up the bottom half of the carb and will reinstall and hope for the best, although I don't give it much chance of working.
Thanks again for all the help.

Ahh, yes, the fuel bowl solenoid---HAH!!
A friend and neighbor had a problem with that solenoid, as the black plastic plug in the end of it had melted and burnt off! Andd i got the job of fixing it!
So i removed the solenoid, and found it was melted where the wire plug went into it.
Off to the mower parts store! Look it up in parts book--got price---WHAAATT!! $83 plus tax??? For that little valve??? Yep!
So, back home, did some figgerin', some trials, some praising the guy (read nut) who designed it!
Found, by running a piece of fine wire thru it to hold the pin in an open position,(crossways), the engine would get fuel and run. And, run it does! Would ya call that Yankee Ingenuity? And, it doesn't backfire when i shut it off!

BINGO!!!! I put the carb back together, reinstalled, and fed it some gas into the top of the carb. The third time I tried to start it that way it kept running. So whatever I did worked.
Thanks again for everyone's help.