Refuses to power up.

mmm ...

You're between a rock and a hard place on this one. The problem is power failures tend to create drop outs & surges that kill power supplies and in some cases even the logic board.

So while you can replace the power supply to resolve that, it may also require a new logic board too.

Now lets look at this from a usability perspective: This is a very old system and Apple is no longer maintaining the OS and getting any newer Apps is not going to be possible. Even using it to surf the web is not going to be safe and many sites are now running newer HTML code that your browser can't support as well.

So how much money & time is this system worth to you? Frankly, I think it would be wiser to get a newer system (new or used).

If you're still game, here's the IFIXIT guide you'll need to follow to replace the power supply: iMac G5 20" Model A1145 Power Supply Replacement. And here is the part you'll need: iMac G5/Intel Power Supply - Apple Part #: 661-3780. Lastly, Apple does not service this system any more as well.