I've found it almost impossible to remove a broken screw and leave a threaded hole that you can use but by all means try screw extractors etc. There are even drills with flutes going the wrong way that you run CCW just for extracting screws. The only ones Ive found are here
. Remember to run your drill backwards with these.
There are devices known as Helicoils
that require a hole bigger than the screw. Helicoils are inserted in the oversize hole to give you the original female thread. Drill out the screw very carefully using a center punch to mark the center of the screw.
Helicoils are very good and will be in fact stronger than the original. (Boeing use them for stronger fasteners on new parts). The only snags are expense and there's a smidgin of know how to inserting them, you may have luck finding a local auto repair place that will do the whole job for you for about the price of a kit that you are unlikely to need again