From the start
1) use a Dremel tool fitted with a tungsten bit that is flat on the end like an end mill and flatten the end of the broken item
2) use a lathe centre drill to make enough of a dimple to start drilling
3) use a standard cobalt drill to start drilling out the hole stepping up one size at time
This both makes the hole and heats up the stuck whatever which is handy if corrosion is the problem.
4) swap to left hand cobalt drills used in hammer mode and press down very hard to prevent it bouncing around
About 1 in 5 will will unscrew with the left hand drill .
If this is beyond your skills then ring round the local engineering workshops for one that uses water drills and get them to machine it out for you.
Not cheap but quick and very clean with zero damage to the item.