Q: iPhone 5 won't restore after recovery mode!

I found this on the site for iphone4s, maybe it would be the same for 5s? It worked for my iphone 4s.

You might need to use recovery mode if:

• You can't update or restore because iTunes doesn't recognize your device or says it's "in recovery mode".
• The Apple logo remains onscreen for several minutes with no progress bar.
• The Connect to iTunes screen appears.

See what to do if the progress bar remains onscreen for several minutes

Use recovery mode

Recovery mode erases your device and restores it, which should resolve the issue. If you previously synced with iTunes or iCloud, you may be able to restore from your backup after recovery.

1. Turn off your device. If you can't turn it off, press and hold the Sleep/Wake and Home buttons at the same time and wait a few seconds for it to turn off.
2. Plug the device's USB cable into your computer only.
3. Hold down the device's Home button as you connect the USB cable to it.
4. When you see the Connect to iTunes screen, release the Home button. If you don't see this screen, try steps 1 through 3 one more time.
   iTunes should open and display a message such as: "iTunes has detected an iPhone in recovery mode. You must restore this iPhone before it can be used with iTunes."
5. Use iTunes to restore your device.

If you don't see your device in iTunes, follow these steps for Mac or these for Windows.

Additional Information

If you started your device in recovery mode by mistake, restart it to exit recovery mode. Or you can just wait—after 15 minutes the device will exit recovery mode by itself.