A power supply is a trivial if the computer is worth saving.
Do an easy test.
Assuming that you're comfortable opening the case and tinkering, find where you plug the power button into the motherboard. It should be a two pin connector or two pins in a larger bus. Unplug it, and short (electrically connect) it with something else. An unused jumper that came with your computer accessories would do the trick. Careful not to short something else...
Report back what happens.