The screen got stuck in a power loop. It would flash the Apple logo for five seconds, then it would disappear for five seconds, over and over. If you pressed a button, the "low battery, charging" symbol would come on, but after five seconds it would go back to its normal routine. I used to work at a SimplyMac, so I knew if this was taken in, they would just try to do an out-of-warranty, full-unit replacement, which is ridiculously expensive for an iPhone 5. Since it's out-of-warranty anyway, I figured I'd give this a shot.
The repair went pretty seamlessly. I did have a bit of trouble getting the old battery to break loose, as I did not have a hair dryer on hand. Afterward, the device powered on no problem. For anyone who has this issue, TRY THIS FIRST. Apple is a brilliant company, but they have many rules that they must follow. This will probably work for almost everyone facing this issue.
Don't be too scared of popping the battery out. I had to be a bit forceful with mine, but they're built not to explode, just expand. You definitely don't want to puncture it, but if you use the tools right, you won't.