Popped the cover off the old power supply from the computer to have a look at the innards. I had blown a lot of the dust out of it earlier, but there was still a bit clumped around the wires that come out the back.
It's pretty densely packed in there. At first glance, all the components on the board still look in pretty good condition. Since the main culprit for why the power supply went flaky was the main power switch, I looked there first.
It will still probably be a fairly easy fix, but I'll have to replace the one corroded wire along with the switch. The other end of the wire isn't going to be that easy to get to to remove and replace. I'll probably leave this for a project to work on after moving into the new house.