On the bright side, at least that model has an AGP slot, which is more than you can say for many off-the-shelf Happy Meal? computers. From the quick-setup sheet
here it appears that it might even accept an ATX power supply, although the front-to-back depth of a full-sized ATX power supply could collide with the rear of your optical drives. But there's at least modest hope that you could install a new PSU and get away with your upgrade.
Another option would be to pick up a decent new case & PSU, maybe an Antec Sonata with its stock TruePower 380 or something, and throw your HP stuff into it.