Try something like this.
https://www.buyspeedout.com/?MID=4878676There are other brands out there like this though. I think Sears has their own. Irwin has thier own.
For the future, when trying to get screws out, I'll second the impact gun idea.
I have a small 12v Rockwell impmact that takes 1/4" bits, It's amazing what that thing can get out when a screw gun would just strip it.
Chris