So I tried the usual stuff, making sure it was getting power, disconnecting and reconnecting the supply, pulling every peripheral, yanking the HDD and memory and reseating them, even reseating the power supply connection. I went to dell.com to see if there was a solution, and they suggested I try everything I'd just tried. I checked the forum, and it appears that other people with this model have had this problem. So there's this board with the power button and it has problems sometimes, but someone knew of a brilliant method of fixing this. MacGyver would be proud, for it involves a paper clip. Yes, just jam a paper clip from the green wire to any black wire on the power supply connection to the motherboard and voila, the power button works again. (Okay, so I used an actual short piece of insulated wire, but you get the idea.)
That only took most of two hours. On the plus side, my heat sink and fan are cleaner than they've been in years!