Hello,
We replaced the shorted-out power supply with an identical replacement to what was in the computer.
Now there is a slow beep, the lights come on, but nothing else happens
Sorry it was MB. The "slow beep" is the BIOS telling what part failed "post on self test" = POST. The beep codes are bios specific, so you'd need your manual or the name of teh bios (e.g. ami) to decode.