Interesting approach.
Put a nut over the hole the bit is in and build up a weld up off the bit and out of the hole and weld it to the nut then back it out with a wrench. Tig would be best but it can be done with a mig also. I have done this technique several times on bits, bolts, ez outs etc. Never once has it failed me.
You can do the same technique on the bolt that's broke off. The heat from welding will loosen it up and it too will come out. The whole process takes about 10 min. As soon as the weld is not cherry red, start going after it with your wrench. Do not let it cool all the way off.
IIRC .. In a post not long ago,, a member swore by using a carbide(?) masonary bit..
hope it helps..