How can I remove a recessed bolt that has the ball from the tip of an allen wrench broken off in the head?

You'll need a really good drill bit and an easy-out. Buy an easy-out with the correct size drill for the bolt you want to remove. Drill the hole in the shaft of the bolt and twist the easy-out into the hole. The easy-out will be left threaded, so screw it in the same direction you would normally be removing any bolt, which is counter-clockwise. It will snug up, then keep turning till the bolt comes loose.