Find yourself a brand new set of good quality left-handed drills. Start with a small diameter, go all the way through the bolt (lubricate! Go slow! Stay centered! Don't break the drill!). Repeat with a slightly larger diameter, and then larger. Do not go larger than half the diameter of the bolt.

At some point the bolt is going to unscrew itself under the action of the drill. If it does not, it's then time to put a bolt extractor in the hole, taping it a few times with a hammer to make it bite and then turning it slowly counter clockwise. Using a bolt extractor is a tricky business, those tools are really strong but brittle, so *go slow*, take your time.

If the bolt extractor doesn't work, continue to drill larger and larger until you can get a tap in to rethread the hole