I'd try what Oren says, including the tapping,
The description of the tool he lists sounds pretty good and gets a lot
of high ratings, plus quite a few low ones. But I've never used it,
have no experience with it.
I'm surprised no one suggested left-handed drill bits. Often when I
use those, while I'm drilling CCW, the screw comes out. Start with a
small bit, even if you have to use a clock-wise one to make the starter
hole, and get gradually bigger, but no bigger than the hole! (You can
see how big the hole is by looking at a new part at a hardware store. I
think they're all the same in a given country.) Home Depot has a
couple sizes of left handed bits by a famous maker and the only other
place I know that has them is Harbor Frieght, which sells a set of four
different sizes.
Oren's device is meant to run in a drill that's going counter clockwise
too. That's a good sign.
I think what all of us shoudl do is go around our homes loosening and
retightening every machine screw in the house, so we don't get stuck
like this. Starting.... NOW!