Lets get that phone working kwaldron! I may have the fix for you. Hold down the Home and Sleep/Wake button until the Apple appears and then release. Once the device has started up or displays a "connect to iTunes" screen, try to restore again. Let us know if the issue persists.
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Try two steps below to fix iPhone 5 won't trun on problem:
1. Hold the power and home/circle buttons for 10 seconds to perform a Hard Reset.If that doesn't work, connect your phone to a computer with iTunes (where you back up your data).
2. Hold the power button for 10 seconds, then let go. Then hold the home/circle button AND the power button together for 10 seconds. Let go of the power button and keep on holding the home button. Your iPhone should display a white screen or your computer will prompt you and tell you that your phone is in recovery mode and to perform a hard restore (DFU Mode). In both cases, you should be able to restore the phone with your backed up data.
Also find the instruction on this article