my iphone5 is shutting down and has no power to turn on what do i do?
first thing I would do is restore it back to factory one more time.  If that doesn't work, call apple and tell them, they may give you a credit, and lastly try to trade it in and get the iPhone 6 when it comes out, or if you can't wait till then, try to trade it in now.  As far as fixing the issue,it sounds like you may just have a lemon. You could tell them at an Apple store that they said this would fix it, and it isn't working, they should have given you a new phone instead of trying to replace the battery.  But next time you get a phone that's a dud, or any electronic, request a replacement, say its no good.