I would not even try to cut in to the K&T or add to the circuit at all. A dishwasher should usually be on a separate 15 or 20 amp circuit, I would maybe think about putting it on with the disposer if you were aware not to use them together and they were on more modern wiring. Highly suggest that you leave the K&T alone since K&T is most dangerous when it has been modified incorrectly or overloaded. Instead go get some 12/2 romex cable and pull a new circuit to the new dishwasher from your breaker panel. If your home still is using a fuse box then you need to hold off on the dishwasher project and get a new breaker panel installed first. if your homes K&t wiring is ok and not damaged or unsafely modified your electrician can just put all the k&t circuits on 15 amp breakers if your local code allows k&t to be still in use. I am in an older home that still had some K&t when I bought it. now I have it down to only a few circuits and those are mostly just running lighting and ceiling fans that were in the house. I had to add circuits to the upper levels. I am adding a dishwasher my self very soon too and I will just put a new beaker in for it.

Pretty easy to replace old with a non-lighted one - check youtube for a video. Remember black wire goes to gold terminal, white to silver. Obviously turn off the breaker first (had to say this). Wires up the same.

@Lisa Cain. Thank you so much for your quick response! I think I am going to go with the raised button. I have the recessed button in my basement bar and it doesn't get a lot of use, but the one on the kitchen counter will get tons of abuse with my family. Do you know if yours is the WestBrass brand? My kitchen remodel is nearing completion and this is one of the last things I need to decide! I am looking forward to the day when everything is installed and all the decision-making ends.

Definitely on the counter. My sister has one under the sink where you have to open the cabinet door to use it, which is a big PITA because you have to step back, open the cabinet door, press the air switch, wait until you are done with the disposal, press the switch, then close the cabinet. The only thing I would add about installing the air switch in the counter is that you want it to be installed at the end of the line, near the corner. Some people put the air switch right next to the faucet which you shouldn’t do. You don’t want it too near the faucet handle. Because you don’t want drippy hands at the faucet handle constantly dripping onto the air switch. Here is how it should be placed: