Just a piece of advice from an auto tech who has seen many broken bolts, if the screw or bolt was cross threaded while tightening and the head broke off, don't use an extractor, odds are it will break then you will have a piece of hardened steel embedded in your screw (not fun). The left handed drill bit is the first thing I would try, making sure you are PERFECTLY centered, if it doesn't come out, then it's time to get a tap of the same size and tap new threads, sometimes you may have to go bigger and/or a different pitch.
Good luck