Did your iPhone 5 battery run out yet? Is it stuck on low battery mode & won't charge? I had a similar issue with my 4S & I went to Apple's genius bar. They don't take in walk-ins I believe. Funny thing is the guy that was helping me out told me my phone had a 2001 error & was a hardware issue. My warranty was up so the only option was the replace my phone. Later on I took it to get it checked up by a well iPhone expert of some sort who charges to fix phone problems. Amazingly, he fixed my problem.
Cherry on top, a replacement battery fixed my issue. Battery = 50.00 something, new phone = 500.00 something. Total bs. They wanted me to replace my phone for only a battery issue. My phone's charging now again and has been working for 2 days already since I replaced the battery.