My first thought is...I think that there's almost no way that the keyboards space bar is getting warm.
There's really not enough electricity flowing thru the 512's keyboard cable to make anything warm...and if we were to disassemble the keyboard...you would see that the spacebar itself is so far removed from any electronics (or heat source) that something inside the keyboard would have to be REALLY overheating to get the spacebar warm.
These keyboards are VERY very simple in design...there just isn't much in there to make things warm. But maybe I'm wrong...you're certainly right there experiencing things.
In any case...if the computer is working fine...then the problem probably isn't the logic board (and you don't want it to be)...since there's very little chance of you getting a replacement logic board at this point 27-28 years later.
Most likely (if nothing on the keyboard is working)...the keyboard cable or the keyboard itself may have gone bad...and these two things would be a much better place to start in terms of troubleshooting. Of course you will need parts...gotta check out e-Bay...for folks selling old (but still working keyboard cables & keyboards).
Don't be too shocked when you see the prices!!
- Nick