A/C does not work

A/C slowly stopped putting out cool air. Then it would come and go depending on how hot it was outside. Sometimes if I let it run for awhile, it would eventually get cold. Now nothing but hot air. It took more than five years for all this to happen, but up until about six to twelve months ago it still occasionally blew ice cold air.

Did not need the A/C in Washington state. Now I live in Los Angeles area. Problem started happening about two years after I moved to California, 2009.

Three mechanics have looked at it recently. The compressor works. No Freon leaks. They think it is electrical. Replaced the computer. Still does not work. New mechanic is painstakingly tracking backward to see where the problem is. He thinks it may need a thermo switch (see attached photo). Something about telling the A/C when to blow cold air?

Any ideas about what is wrong or where I can get parts?