The first thing to do is buy or make a bushing to go into the hole. The bushing will keep the drill centered. It is very hard to drill out a steel screw when the screw is in a softer material. The clearance hole diameter for a #6 screw is 0.146. You can make a drill bushing in the following way:
Take a piece of rod that is about the same diameter as the screw hole, or slightly larger. Cut a small piece, about an inch long. You can use a #8 machine screw for this which you can buy at Ace Hardware or Home Depot. Cut head off of this screw and chuck it in a drill press. Turn the drill press on and use a flat file to reduce the OD of the screw to fit into the #6 hole.
Put a 1/8" drill in a vise with the drill pointing up. With the drill press running, and the #8 screw in the drill press, you should be able to drill a hole into the #8 screw sith the 1/8" drill and the hole will be close to center. I have done this when no lathe was available.
When yo have a hole about 1/4" into the #8 screw, take the screw out of the drill press and cut off the end where the hole is. Put this small bushing into the hole where the broken screw is and drill a hole in the broken screw. You can then use an easy out or left handed drill to remove the broken screw.
If you cannot make the bushing, or cannot find someone to help you, you could have a machine shop make the bushing for you. I am sure it wouldn't cost much as the shop can make the part in about 10 minutes with a lathe.