Patrolman previously posted:
"it is out, drill out the rest of the head so you can remove the bracket. Then you can hopefully apply good penetrant and hopefully remove the rest of the screw by grabbing what remains with some vise grips."
I am with Jeff here: do what it takes to get the extractor bit out - I would probably go the dremel grinding route. You could even grind the entire head off (I first bought my dremel 20+ yrs ago needing to grind the head of a CJ7 steering box bolt that was stuck and I trashed) - then the bracket should come off and you can work with what is left. Hopefully a decent amount of thread will be sticking out that can be worked with more easily - either vice grips, welding a nut to it, etc. I would also then try to head up the threads - it helps loosen the loc-tite. There is also a product called 'freeze off' (or something like that - comes in a blue spray can). It helps shock the item and loosens it up. Plus then any light scuffing or damage to the surface would be hidden under the bracket when its put back on.
I have fought a number of stuck bolts and twisted off heads, rounded out screw heads etc - usually comes with the territory on rusty old jeeps. Someone once told me it was 'character building' (not that I believe that). For future reference, as Brian mentioned above - especially for screw heads (allen, torx, phillips etc) - an manual impact screwdriver goes a long way and helps avoid the situation entirely, especially if you put some pre-treatment penetrating oil, shock, heat, freeze off etc on it first. I've got a rounded out, recessed allen head drain plug on the front diff of the tacoma - its probably never coming out - l just change the fluid now by pumping the oil out and refilling. Good luck!