For dealing with broken bolts, round heads etc.. a lot of these happen in the automotive part. (and I've got a lot of experience with these as an automotive mechanic in the past and a lot of wrenching on all sort of things since then)
Anyway here is my order for solving this thing:
0. Add penetrating oil
(pb blaster/liquid wrench/etc)
1. try with normal wrench
2. try with impact wrench
2'. add heat from a heatgun (or a blowtorch) to loosen the paste that usually exists on the threads (almost all automotive screws have some loctite antiseize/grease/paste of some sort to prevent loosening, to be airtight, etc.. that thing is useful for unscrewing by heating the damn thing)
3. (if it's an INBUS bolt, (allen bolt), jam with a hammer the next TORX bit
4.a) drill a hole and use an extractor
4.b) weld something (a nut, another bolt, etc) on the remaining head/broken bolt (works if the base is not weldable, like a broken bolt in an aluminium engine)
4.c) same thing but with glue - yet if a screw extractor failed, any glue is a lot weaker and i don't use glueing. it's messy requires time to set, etc..
5. Drill carefully trough the bolt not to touch the threads - then use a tap to chase the threads.
6. If you messed the threads, enlarge the threads and use helicoil
7. If helicoil not available for the required application, enlarge the hole bigger and use a larger bolt, or insert a reduction bushing.
8. reposition the hole elsewhere, re-engineer and make a different mounting, cut a large hole and path it with new material welded and tapped accordingly or if anything is not possible (like complex cast part like engine/transmission blocks) then it's a loss and have to be replaced.
End point is that you are at step 3. use it! I've encountered many stubborn inbus bolts (allen) that responded right away with a torx bit/impact wrench and some heat (or just the torx bit jammed is enough)
So the order to follow should be from 0 to the end.