You noted that it had a hard drop ?? how hard what did i land on ??
you noted that you also tried reloading/restoring from itunes ??
if thats the case it sounds like either board level solder damage and or that drop
could have damage processor or ram>>>
have your tried this
Hold both the home and power buttons at the same time. When the Apple boot logo appears, let go of the power button but continue holding the home button.
Connect your device to your computer and open iTunes. If you don't have a computer, borrow one from a friend or go to an Apple Retail Store or Apple Authorized Service Provider for help.
While your device is connected, force restart it: Press and hold both the Sleep/Wake and Home button for at least 10 seconds, and don't release when you see the Apple logo. Keep holding until you see the recovery mode screen.
When you see the option to Restore or Update, choose Update. iTunes will try to reinstall iOS without erasing your data. Wait while iTunes downloads the software for your device.
If the download takes more than 15 minutes and your device exits recovery mode, you'll need to repeat these steps and choose Restore instead of Update when you get back to this step.
After the Update or Restore completes, set up your device. If you previously backed up your device with iTunes or iCloud and had to restore your device in step 3, you might be able to restore from your backup during setup.