Computer fans are 12V BLDC motors, this thing is an AC squirrel-cage motor. They have totally different supply and control requirements.
See the two surface-mount black things above and below the capacitor that's overheating? My guess is that the top one (3 pins) is a triac which is doing phase control on the motor. Replacing that on the stirrer that doesn't work at all might get you somewhere, or it might not.
If you choose to use a simple lamp dimmer for speed control, it may have a tendency to wander with varying loads. This is not much of an problem for quick-mixing chems, but for slower, longer-term mixing, wandering speed can become an issue.
I will try to get some better images tomorrow. I may have a guy that an look at it in person.
If the part has thermal paste, it's not a capacitor, what's the label for that part on the board? like c??, r??, q??. If the part that gets hot has become disconnected from the heat sink it could have been destroyed by the heat.
The part with thermal paste is not the part that gets hot. In the photo I attached earlier, the part I circled is the one that gets hot, it is labelled "C2". The part with thermal paste is at the top of the board, and the paste can be seen as a grey goop on its surface.
Is there an electronic engineering department in the university? (look for a lab technician rather than an engineering student - they won't have a clue!).
There is, but virtually everyone is absent right now since it is the summer. Most people aren't back until late August.
C2 is a capacitor. If it's getting hot, something else is causing it to get hot. You'd need a tech to figure that out.