Disconnect as many internal devices as you can. Power up with a minimal hardware configuration. So remove HD, remove CD/DVD, airport, RAM (unless you need one stick to detect the chime), remove the graphics card. So the idea is to minimize the config to isolate to a particular internal device.
If power remains on with these devices disconnected look for any debris in the connection points, RAM slots, PCI slots, anything and clean them up. Clean anything that looks dirty or corroded. Scan the logic board with your eyes, looking for anything corroded or suspicious.
So if you suspect power, with few devices attached you might see that it remains on. Add back only one internal device at a time and power up. Any change? Keep adding back parts one by one and observe if the issue is reproducible or not.
Be sure to clean up any debris in the machine that might be conductive causing a short as indicated by Demik.
Could be the PSU but then again with fewer devices requiring power it might make an already weak PSU (if it is the case) to run perhaps more efficiently getting the system far enough to witness some desired result.
Doesn't the quicksilver have a few LEDs on the logic board that can tell you something while the machine has power?
Keep us posted.