Article: Find a nut that fits snugly around the screw head. Bond them together using a two-part metal-to-metal epoxy, often sold as "weld bond." Wait for the epoxy to cure as directed on the label, then grip the nut with a socket wrench and rotate. If you don't have a nut the right size, you can adhere a smaller nut to the top of the screw head. This will not provide as much leverage. Breaking the screw usually relieves pressure on the screw shaft, making it easier to pull out but if it doesn't work, you've eliminated most other options. Choose a drill bit slightly larger than the shaft of the screw, so the head detaches completely when you drill through. Start with a center punch to make a hole in the exact center of the screw, and take care to drill directly through the center. Once the screw head breaks off, grip the screw shaft with locking pliers and rotate counter-clockwise to remove. If the screw head isn't flat, file it down or grind it with a dremel and a pointed grinding stone attachment. Center punch and drill once you have a flat surface to work with. If all else fails, hire a machine shop to remove the screw using electrical discharge machining (EDM). This may be your best option if you used a screw extractor that broke off inside the screw.

What is a summary?
Attach a nut to the screw with epoxy. Drill out the screw head. Hire a professional.