Can you post a picture of the matter or link? loger

I ideal situation would be to drill small hole in the center of the broken bolt. then set up to just under 1/4. The tap the threads cleaning out the old shank without heili-coiling it. The problem with it being broken of down inside the hole you can grind it flush and get square flat surface to find the center. If not flat usually the drill bit will go off center. Either way try to drill the piece out first and use a pencil grinder to remove any off center metal so you can try to tap the little bit of remaining threads. What ever you do don't break the tap off. If you do it will be twice as hard to get that out. Use plenty of lub. and go slow clockwise then counter clockwise. The heil-coil kit will have a larger drill bit to so you can install the heil-coil and still have 1/4" threads when the heil-coil is installed

Options: (drilling dead center is your challenge)
1. Drill out very close to the threads and you can usually get the remainder with needle nose or pick.

2. Drill 5/32" and use an extractor (which are that small.

3. Drill with a reversed drill bit in reverse and it usually comes out (lightly oiled).

4. The helicoil would be my last option work wise.