Look for a shorted tantalum cap somewhere (usually looks like little round yellow or orange balls of plastic on two wires). First remove all the expansion cards and see if the system turns on without them. If so, then put the cards back in, one at a time, until you find the one that prevents it from powering up.

If it still doesn't power on without any cards, then the short is on the motherboard - likely a tantalum cap somewhere.

Use a multimeter on the lowest ohms setting and test the resistance. As most of the smoothing caps on a circuit are connected to the same rail, they will all appear shorted when you first test them, but pay close attention to the ohms readout on the multimeter. The one that returns the lowest resistance value is possibly the suspect cap.

Looks like your advice was spot on. Audio daughtercard C20 appears to be shorted, and sl boots fine with daughtercard removed!

Interestingly, it looks like someone rebuilt the audio daughtercard at some point in the past and not very well, so I think they knew there was a problem with it of some sort but never actually fixed it...

ANy idea what type of capacitor this is? 106k seems to be a ten microfarad capacitor, and it's probably not HUGELY important what type I use, but I'm curious as to which type this is regardless.

Those are axial tantalums. They can short sometimes too.
Desolder it and replace with a new cap. I like to replace tantalums with aluminum electrolytics as they're somewhat less likely to violently fail.
On that particular cap, the narrow/conical end is the plus side.