Drilling out a broken exhaust stud is best done with the engine out of the vehicle.
You used to be able to buy a drill fixture to drill out exhaust studs. I don't know if they are still available. You could ask the guys at AirSpeed.
You will have to unbolt the exhaust system from the head, to drill out the broken stud. File the end of the stud flat and center punch it. Use your drill fixture to drill it out. If you can't get a drill fixture, you can drill a small hole down the center of the stud, and progressively get bigger, until you get to the tap drill size. (6.75mm for M8 stud) Retap the hole when you get to 6.75mm. Use sharp tools and lots of lube. It also helps if you use a left hand drill bit to do this job. These can be difficult to find, but they make the job easier. Sometimes the left hand drill willl lock in the stud, and simply unscrew it. But this does not happen very often with exhaust studs, as they usually rust themselves into the head.
There are also oversize studs available, if you need to make an oversize hole, M10 on one end, and M8 on the other.
Do NOT use an \"easy out\", or stud extractor! They rarely work, and break easily. And when you break it off in your already broken stud, then you are really \"F\"ed. It will be very expensive to remove a broken easy out.