Thanks to the suggestions here, I was able to fix my wife's "broken" iPhone 5. She recently dropped it while in a case. It didn't crack or suffer any obvious damage, but it refused to turn on without the charger connected. We tried to do the iTunes update and restore but neither worked. It would go to the Recovery screen but never beyond.
After reading about broken batteries, I opened it up and discovered... bad quality control from the manufacturer. It was missing 2 screws from the upper cable bracket, but more importantly, it was missing the entire bracket that secures the battery connector in place. This cable had popped out when it hit the floor so it wouldn't power up on its own. I reconnected it and reassembled it and it came up right away.
After doing an update through iTunes, everything was back to normal. Now, I am a hero! The battery idea made sense, but I expected to see it broken. Unplugged was easier to fix and cheaper.