It sounds like something may have gone wrong with one of your USB ports somewhere. Do you have any cards installed that add additional USB ports? If so, remove those and see if the problem persists.
Otherwise, check inside each of your USB ports and make sure there is nothing stuck inside any of them or that none are damaged inside. If everything looks good, blow them out with some compressed air. The next thing you can try
resetting the System Management Controller, and maybe
NVRAM.
If the problem persists after this, it's likely just a bad USB port somewhere on your machine that is actually drawing too much power. If it is a problem with the front USB ports, those aren't too terribly difficult to replace. I believe you can remove the front-facing ports and power button and power on the machine with a pad on the logic board, though it's been a few years since I've worked on one of these, and I can't find a page to reference where that would be. If it is one of the back built-in ports, however, that would be a problem with your logic board, unfortunately.
Edit: Found an Apple repair guide for this machine
here on iFixit. Pages 29-31 describe what to jump on the machine to power it on from the MLB, if you want to test with the power button disconnected.
