I have often used from a set of short shanked left-hand drill bits to get a bite on a broken bolt, trick I use is to drill a short ways with a size slightly smaller than the LH bit, then while bearing down and using a reversible drill (my ONLY drill to use is the Milwaukee right angle drill) slowly start up and while drilling in revers the flukes dig in and you can thread out the broken bolt providing its not too corroded.

That Milwaukee 3/8's angle drill is my absolute favorite.