Presumably sheared when an over long screw bottomed in the hole.
No chance of being able to access the stub with a small cordless drill, to start opening up towards tapping size. (Obviously the need is to get close as possible to the centre point )
Minimise, as much as possible, the amount of disassembly, so try to remove the broken stub with everything in situ. the more that you take apart, the more that you have to put back again, and adjust accurately.
You could possibly try using a left hand drill on the sheared off screw. With a bit of luck, it will bite in hard enough to start the stub unscrewing. It it does, it can be unscrewed by hand or with a pair of pliers. Being 4BA, you will need a small drill of course.
No chance of being able to access it with a small centre punch and start unscrewing it?
Another remote possibility might be to make up a tiny chisel, (4BA tapping size or less)and to try rotating it anti clockwise whilst hitting it on the end with a small hammer. It would be nice if the tool bit in and the rotation combined with the shocks, started the screw moving..
Even if you can manage to drill down the middle of the screw remains, do not use an "Easyout", even if you can find one small enough. Unless you are lucky, this is likely to expand the remains and jam it even tighter. A friend of mine from the Toolroom described Easyouts as the most misnamed tool!
Howard