If Gateway's pinout info for the PSU is accurate, then it appears to
not be a standard ATX pinout on the 20-pin power cable. I compared their
diagram to an official ATX specsheet. They don't match up, unless Gateway has goofed their diagram up. You could do a reality check on that using a live Gateway, a live standard ATX system, and a voltmeter.
If the affected computer simply has some data that you need, then here are a couple of other ideas:
1) take the hard drive out of the Gateway, put it in a working computer and recover the data to the working computer
2) get out an idle computer, put the Gatway's hard drive into it, and do a Repair-install of Windows. To do this, start Windows Setup from CD, and the first time it asks if you want to Repair or do a fresh install, choose a fresh install. It will get ready, show you the EULA, then look at your hard drives and you should have it install to the existing C: partition. It will notice the existing Windows installation and ask if you want it repaired, and
now you say Repair oh yes pleeeze pleeeze repair it

and just let it go through the motions of installation. It should come to the surface with the programs and data intact.
Good luck
PS: method #2 will leave you with a "raw" Windows installation that needs patching, so run it by Windows Update at the earliest opportunity.