Have you checked the serial number of your iPhone 5 on Apple's iPhone 5 Battery Replacement Program page?
As shown below, mine is one of the unlucky ones... No wonder some time last week it suddenly dropped from about 80% to 0 and I thought it was dead! Have to wait for a long time for it to be charged.
This is not the first time my phone has issue, remember the previous sleep/awake button replacement program? Mine was in there too....
My iPhone 5 was replaced, see this post for details.