your reply is exactly what is required. the drill must be in the center of the broken stud . use of a piece of tubing of any metal copper is very good to ensure you bore into the center of the stud. using the drill to do the work ,,,, lightly work the bit to bore into the stud. using a slow speed with the torque limiter on not too high. as you cut into this stud I expect the stud to begin to unscrew.
If it does not un screw then you drill until you go threw the stud you will suddenly feel the bit drop . stop drilling then remove the head and using an easy out remove the stud.
I have used this method on bolts much larger on very large vehicles . to correct this type issue.
never use chinese bolts and or reuse bolts torqued to the max allowed . this is why I had to do this type of repair. the shop did not have new studs. reusing old studs under heat and stress is why this occured for me.