A few things. Last year to raise some extra cash I recycled computers for precious metals. I tore up many, many towers. I'd say about 10-20% of the computers I ended up with had bad power supplies and many that I ripped apart had components that had obviously burned up (giant charred spots, ash, etc). With the amount of juice pushing through them I'd be very hesitant to do any sort of homemade wiring job to them.
If you find the model number of your power supply you can probably find one used online for cheap, often on eBay. Many of the power supplies I was pulling were being offered on ebay for $5-10 and I wouldn't be surprised if you could find a used but working one pretty cheap.
Fun fact, before I try anything else with a tower that will not turn on I pick it up about an inch off the ground and drop it. (With the power off that is, don't do this with it on as it can ruin the harddrive). You'd be surprised how often this can fix things.