I think we have enough to call this an answer now. Here's how to do this:
If you can't use an existing hole, insure that there is clearance on both
sides of the firewall before drilling. You can use whatever bit that fits. Although if you have limited space and can't use a prick punch make a dimple so the bit doesn't wander, a step drill might be more ...
If you have a welder.
Weld a metal rod to the top of the drill bit at a 90 degree angle from the drill bit.
You can then use the rod as a lever to turn out the drill bit in reverse. You may want to take a hammer and punch and hammer on the end of the drill bit a few times to loosen it.
How much of that drill bit sticking out?
IMO opinion, even if it a ...
The drill bit has flutes (grooves in the sides). Try driving nails into both flutes, then grab with locking pliers and turn. Lubricate with penetrating oil first to help.
On second thought- instead of nails, which are generally very soft steel, use tougher steel pins... the easiest of which to find is the shank of a dull drill bit, everyone has some dull 1/...
Most internal plastic in motor vehicles is ABS, which is relatively "soft" when it comes to drilling. A conventionally ground bit can and will grab during the drilling, but it's unlikely to crack the plastic. Harder plastic such as acrylic should be drilled with special bits or special techniques such as reversing the drill motor, but ABS is pretty ...
If the bolt is just needed to keep the brake cable from dangling around and has absolutely no critical function then I'd try to clamp the bracket (perhaps by some U-bolt) to the control arm and forget about the broken bolt.
Regarding your request for drilling:
Your stated scenario indicates that you experienced something called "work hardening". ...
Thanks to everyone for your input! Well I finally got the wheel off. After MULTIPLE cobolt bits I managed to drill down on three sides (triangle) around the broken bit and was finally able to break out the center part with the broken bit. Man I do not ever want to have to do that again. These lug nuts must have been hammered on by a 1000 pound gorilla ...
For go the previous answers! I used to do it all the time as a Big truck Mechanic. use two lite hammers and two punches half the diameter of the drill flutes(grooves on side of the "drill bit"). Ask a friend to operate one set of hammer and punch you the other set. using the drill flutes You both set punch at opposite directions,close to 90degree angle to ...
If the bit isn't well jammed into the hole you might be able to get the tips of some needle nose pliers down the flutes and grab hold of it with that. If you can get hold of it, unscrew it out of the hole. This has worked for me but the bit in question wasn't tight in the hole.
Ideally, with a heavy duty screw/bolt extractor. If that does not work, here is a helpful walk through I recently came across about how to remove a broken but from an engine block:
For under $100 US Harbor Freight sells a locking lug nut removal tool set. It is a set of sockets that cut into the damaged nut and allow it to be removed. The advantage of the set over individual sockets is you can use progressively smaller sockets if the lug nut continues to round off.
In theory you could try a solid carbide, perhaps even an diamond-tipped, drill bit. Attention: Those tips are probably more expensive than a tow truck to the garage, success is not sure, and need careful handling.
In practice it would be more viable, as Solar Mike commented, to remove the hub and replace it / send it to a garage.