I've been meaning to post this before, but after breaking off a bolt today I figured it was time.
I was working on my old Austin gasser today, trying to fix a slow leak around the speedo cable outlet. It's got a TH350 behind the 327, and as you may know the "bullet" that the speedo screws into has a Oring, with a 1/4-20 bolt holding it into the tail housing.
After replacing the Oring I put the bolt back on and tightened snug. At that point the bolt snapped off and the tab, bolt head and lock washer all fell into my catch pan. Of course I freaked out at first, but then settled down to the task of removing the stuck bolt which was less than flush with the aluminum housing.
I pulled the housing to take it to the work bench to make it easier. Removing bolts that are steel and in a soft metal like aluminum makes things even worse. I start by getting the Dremel out and using a round head burr to grind the bolt smooth so the bit wont "walk" into the soft metal around it.
Once I got the bolt smooth I got out my drill and drill index. Pay close attention here! My drill index for removing stuck bolts and screws is a set of left hand twist bits! Left hand bits are key to getting bolts out easier, as the left hand twist doesn't further tighten the bolt into the hole as it drills. A reversible drill motor is needed to run the bits reverse, as they cut CCW, not CW like normal bits.
With a small indent ground into the broken bolt or screw I can easily start the drill bit centered, so now it's a case of just drilling the bolt with a bit a couple sizes smaller than the bolt. If you're lucky the left hand bit often catches the bolt and spins it right out when it does. That's what happened today on mine, so it made life simpler. If not, then once the hole is drilled deep enough, get an easy out and insert it into the hole.
Some prefer the pound in easy outs, but I like the screw type and depending again on the size I either use my small or large tap handle to be able to push it as I turn. With the proper sized hole the easy out will easily grab the bolt, but if it's too small it wont grab and you'll need to drill it out a little more. If you do drill one out larger be very careful! Drilling a hole in an existing hole will cause the bit to grab, and if it breaks you've really complicated things even more!
Hope this helps any of you guys who end up in this unfortunate situation!

 

Left hand drills get 90% of the broken off bolts I have done. I have a lot of people that bring me that type of stuff. Drill press works a lot better then the hand drill. If rust or oxidation is the issue a little heat down the hole you just drilled and some water real quick before you use the easy out can help break the bond and harden the bolt a little so the easy out sticks better.

The worst bolts are the steel bolt in aluminum threads. Without and treatment these things can fuse together. Lots of heat. Aluminum and steel grow at much different rates, In many cases Heli coils is the best fix for that.

 

The one I really hate is when a hardned bolt, which is too hard for normal drill to cut, gets snapped off. Luckly there a machine shop supply place nearby, and they carry solid carbide drills. They are very hard, and expensive, and need to be cooled while working otherwise things start to melt.

 

Yeah the drill press is always my favorite, but with an odd shape like the tail housing it wasn't feasable to do it with my drill press. I've used my mill also, if I can secure everything well.