Howdy, welcome to techIMO!
The old PC - does it have any fans in it (including power supply fan). If so, do any fans come on when you press the power button? Any lights flash anywhere?
Is there a switch on the power supply, and is it on?
Not sure what else we can do unless you can give more details of the old PC - CPU, motherboard, etc.
For securely removing files, the freeware
SuperShredder works well.
Cheers
Mick