iPhone 5 won't start after an iTunes recovery
Force it into Recovery Mode, then restore... 


Switch off the device &amp; leave it off.
Connect the USB cable to a computer with iTunes [ideally one that has been used with the device previously]
Whilst holding down the Home button, connect the device to the computer [keep holding Home until you see the 'connect to iTunes' screen.
iTunes should launch &amp; detect the device is in Recovery Mode.
Use iTunes to restore.


Ref: If you can't update or restore your iPhone, iPad, or iPod touch