FWIW, I used a hand drill to drill mine out. That way you've got more control over speed and not going too deep, although perhaps I've more experience using it than other people, since you do need to hold it pretty still vertically while turning the handle so to avoid drilling through the threads and into the plastic. You may be better off with a variable speed dremel type thing which you can just hold still while it works.
Like I said earlier, gauge the depth of the pit before you start and wrap a bit of tape around the drill bit so you can see if you're going over depth. If you used the right diameter of bit then, you should have removed most of the body of the screw, but not touched the threads, and not carried on past the end of the screw, so that you can pull out the remains with a toothpick, and a bit of patience.