Wayne,
Yes, the suggestion of replacing the heatsink compound is probably the most promising if having a go at a rescue.
Thinking harder about this though, maybe we are at the point of get the thing replaced with something with a reasonable life expectancy while I am still able to do the transfer relatively easily.... A couple years ago, before going into hospital, I replaced someone elses machine with a reasonably high end Core 2, silent graphics card, good well ventilated case, just in case, and it has lasted and will continue to a while longer. He even did a few updates on Debian stable, and escaped unscathed, which was encouraging.
When we get to this point in life, you spend a bit more, but you do get peace of mind in exchange.
I am thinking of one of two alternatives. One would be a Novatch i3 mainboard bundle, package it with a Silverstone cube case, and psu and fans from QuietPC. The other would be just buy the Novatech sff i3. A charity I work with got one of these, and they are reasonably small, light, pretty quiet (though the Silverstone might be quieter and better ventilated given the right choice of components). Not having to install it all is worth something.
Cheers,
Peter