I have seen them before, usually time to abandon the computer when they use proprietary parts. Should be able to plug a GTX750Ti into practically any computer without modification. They are very low power GPUs.
You can try looking for adapters, probably a few out there. I've been trying for a while, but can't seem to find any that look right. Several older HP and Dell adapters though.
If it has difficulty turning on with the stock power supply card installed, read below:
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If you have a voltmeter, patience, and care to take the risk, you can try some re-wiring.
Put the old one back in. One of the black wires is likely a ground. One will be the standby 5V+ those will have to match exactly. Should be supplying 5V when the power supply is plugged in, but the computer isn't running. Turn it on and measure the remaining pins. Write down the voltage for each pin.
You can find the standard pinout for ATX here:
http://en.wikipedia.org/wiki/ATX
Pin removal is quite simple, plenty of Youtube videos if you need a hint. Where they used a very thick wire, you may need to double up. Ideally this would require soldering and/or new pins. But you can just splice them together. Basically substitute voltage for voltage and try it out. No guarantees.