OK, Fisher Scientific is probably worth fixing. One more thing that I should have said at the start: be VERY VERY CAREFUL if this thing is open and you apply power. IT CAN KILL YOU.
The circled part is a low voltage DC capacitor and it shouldn't be getting to 140F. But that's probably a result, not the cause of your problem.
The heating coil might be independent of everything but I wouldn't assume that disconnecting it won't affect other stuff on that circuit board.
You can try Googling for a schematic with the exact model number, it's a slim hope, but you might get lucky. I would pull the thing apart and try to reverse engineer the AC control side of the board and figure out whether the DC supply is still working. Those are the two most likely problem areas. My guess is it might have gotten zapped -- this is a risk for anything that stays plugged in all the time. A second possibility is that something overheated.
The white things by your thumb look to be opto couplers, used to isolate logic circuits from AC circuits. The bent over part with heatsink goop on it just to the right is probably the Triac that controls motor speed or hotplate temp...there should be a second one somewhere...again, can't tell how the hotplate is controlled -- is it a separate, independent control, or does it connect to that board as well?
If you can get a good clean shot of both sides of the circuit board, and then annotate what connects to each of those wires, I might be able to do this by remote control. You can PM me, no need to clutter up this thread until we figure out what's wrong.