I have a Kenmore freestanding 10689482948 ice maker. I had to replace the motherboard. The compressor would not come on. I made a jumper cord and the compressor now comes on and makes ice. The problem is the compressor will not turn off.
As I recall, that was a old hot wire machine. How does it sense the bin full, I suspect it's a old style mechanical thermostat. You will probably find it open. This was a common problem with all the small under counters and replacing it will be a bit difficult. If you can even obtain one. I have a Scotsman that is the same thing in the yard because there was none available. Hope you have better luck with Kenmore.
It has to be a input problem that prevents the board relay from energizing the compressor. Are you sure your not in the wash instead of freeze? Is there a water sensor? Often they also have a water temp sensor that has a anti slush function.
Have you got a copy of the wiring diagram you can post?
Often you can also find the real manufacturer on the bottom of the tag. Kenmore is just a brand name. There equipment is made by others. Refrigerators and stoves mostly by White-Westinghouse. A ice machine will be by one of the ice machine producers. Being a hot wire machine I suspect Scotsman or Kold-draft. But you never know till you see one. Most of the 15 inch under-counters were used in high end domestic use.