Running iPhone side by side with another iPhone
Literally, put the phones side by side &amp; using 2 fingers simultaneously, tap the same part of each screen at the same time, using the working screen as the guide &amp; simply tapping 'blind' onto exactly the same place on the other screen at the same time.  

That way, hopefully, both will be in sync as you tap in the unlock code, then work towards being able to back it up.