First thing I would try is to see if I could jam a size larger allen or torx key into whats left of the hole , to get a grip on it. Between the inch and metric sizes, there are a lot that are close but too small or too big. Sometimes a gentle whack with a hammer, to get the driver to fit is in order. Of course, you want the work secured in a padded vise when you are doing this. You want to have a solid grip on the tool you are trying to drive in, too.
Is the hole a near match for an easy out? I have had best luck with the square shank ones, if just the tip is going to be used. The spiral ones need a hole that goes a fair way into the shank, and usually they are too large for the small screws.
Next is to use a dental burr in a Dremel type tool to carve the head out to be a slot head instead of a hex key socket. Take your time, and be careful, and you can usually crank the screw out with a decent screwdriver. You do want the driver to be as near the width of the head as possible, and as thick as the slot, if you can.
Failing that, the next step would carving or drilling the head off, trying not to drill any deeper than you must. Object is to leave as much stub as you can, if it is required to be able to grab with a small pair of pliers.
Usually, by this point, there is little tension left on the threaded shank that is left, and it will spin out OK.
If not, try carving a slot that you can stick a small flat screwdriver into (CTire has a miniature tool kit that works OK for this stuff. Goes on sale for about $10, has some watchmaker type screwdrivers and small pliers) and putting a little torque on it with a screwdriver.
Drilling down the shank of the thread is about the last option. It can be a right SOB to get the drill to start on center. Dental burrs again, or Dremel bits. Carbide is really nice to have. I like to spot the top of the shank with a ball ended burr, then start drilling with as small a drill as I think will suit. I have, on occasion, made my own easy outs, smaller than were available from the suppliers, by grinding the shanks of broken High Speed Steel cutting tools. Just a tapered square or rectangle cross section that can be tapped into the hole and twisted.
I deal with small, hard, broken screws in very expensive airplane parts almost daily. Escalation of force is key. Use the least that will get the job done. If you do not have the tools or confidence with them to do the job, stop early, and get thee to the pro.
If all else fails, you may have to resort to upsizing the hole, and tapping to a new size, or installing a helicoil, if you have the tools to do so (special tap, insertion tool), or you have a nice guy with a TIG welder fill the hole and you start over with some machining, polishing, and refinishing the mounts. (Uhhgggg!)
Cheers
Trev