Remove broken setscrew in an already damaged hole?
First of all, use a good penetrating oil. I love PB Blaster.

It looks like you started with too large of a drill bit and it walked on you. You need to use a very small Pilot Point Drill Bit at first. Pilot point bits don't tend to wander as much and the smaller the bit, the better it will "grab" the surface. I would start with a 1/64" or 1/16" bit at dead center of the bolt head. Drill about an inch deep. Be very careful to hold your drill straight. If you break the bit off in the hole your going to be in more trouble. Work your way up in drill bit sizes until you reach the size of the hole your easy-out calls for. Use the same concept with your easy-out. Start out small and work your way up. You might have to increase in sizes with the easy-out as well if more torque is required to remove the bolt head.

 

If the bolt snapped off pretty close to the surface you might also be able to do a cross-cut. This is achieved with a hacksaw. You would cut across the surface until you were down into the bolt head far enough that you can use a screwdriver to back out the bolt.