iPhone 5 Battery - Help me diagnose issue
It might not help the OP, but Apple has battery replacement program for iPhone 5. You can check eligibility on Apple site


  Apple has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 devices were sold between September 2012 and January 2013 and fall within a limited serial number range.
  (...)
  
  The program covers affected iPhone 5 batteries for two years after the first retail sale of the unit or until March 1, 2015, whichever provides longer coverage.