Well i f you broke of a screw extractor they can be a real bi%&h to drill, even with a drill press and ideal conditions....Usually it takes a carbide or cobalt drill and lots of pressure. When you say you left the threads, I assume you mean the plug itself left its threads in the cylindr head, and not the lower metal portion of the plug i broken. Big problem with thin sections and screw extractors is that they wedge if its too thin and actually get tighter the more you try and remove ythe broken part if it does not have substantial wall thickness left. If it was just the thrads I would have run a chasing tap in the hole. I think it can probably be remidied as long as you have access...don;t know about the extractor though, sa its just way too much mass to knock into the cylinder and then you would never be able to retrive it without removing the head. Metal shavings are not a problem as a little grease on a stick and some air usually gets most out, so tapping to clean up the threads is usually not a problem, and then just turn it over a bit without any plugs at all in it to allow the movement of the pistons to blow out the trash....If the heads threads are gone, you can always install a helicoil, again no need to remove head....
Tell me more about the broken extractor...........Is it the wedge with left hand sprial, or just a tapered square with flutes on the edges, or ???? HOw much of the extractor itself is in the plug hole and does any of it extend out the the top. Its possible to weld the shank end of a bolt to it, and turn it the opposite way in a lot of cases, but you have to be pretty steady with the welder. Stick or mig or better yet TIG would work. Usually extractors will break free by bumping with a drift punch in the direction (RH) to loosen them, and then finish turning it out with a needle nosed pliers....