This thread may help you. It's long now, but there are lots of ideas, a few schematics, and so on.
It might all be summarized in one phrase: "Replace the capacitors."
https://forum.allaboutcircuits.com/threads/repair-thermoelectric-cooler.101748

Thanks! I'm looking at it now...

 

I'm noticing this thread was started by Soldier Boy. I don't mind helping others, but it would be more helpful if you started your own thread. That way we don't get old facts confused with the new. Last time we heard from Soldier Boy was back in July 2016, so this is considered an old thread.

It would be more useful to you and to us if you started your own thread. Start from the beginning and leave no detail out. That way we don't get confused, and can help you more readily.

A comment on your photographic work: The picture of the circuit side is well done. However, we need to see the whole component side so we can figure out what goes where, and exactly what we're dealing with. Otherwise you could be asking us how to make the fan work and we could be giving you directions to the Grand Banks in the Atlantic Ocean.