I would go very easy on drilling methods, unless you tried everything else first. It's very tricky to drill down the center of a 4-40 or other small screw , and if it goes out the side you are going to ruin the case. I would do everything (heat and Knock 'er' Loose or whatever), then try to spin it out by hand, with an eraser, and if necessary, cut a screw slot using a Dremel cut-off wheel, even if you have to cut into the case a little. That is less damaging than getting off-center with a drill.
If you absolutely have to drill it, for certain, you need to put it into a vertical mill, mill the remaining end off flat, center and dimple it with the smallest center or spotting drill and then drill with a left-hand drill with decent pressure to try to get it to grab and spin out. The problem is that will be a *tiny* drill - Assuming it is 4-40, a tap drill is a #43, you would want to start with maybe a #46 or $47 to cover yourself against getting off-center. That's only 79 thousandths of an inch and extremely prone to deflecting or breaking under the required pressure to get it to "bite" and spin the screw out. At least, cut if off or chuck it up with only about 1/2" protruding to keep the deflection down. If you do end up drilling it out instead of spinning, if you are even a little bit off-center, you are in real trouble because you will never get it centered up again and a few thousandths of an inch means you can't go all the way to the tap drill. Then, you have to try to get a pick to pull up the ends of the threads into a .080 diameter hole.
I think drilling it out completely will come to grief. Worst-case you have to drill it clear and use a heli-coil or keensert. It looks like you have enough meat around it to do that. So trying to machine it out would be my very last choice.
Brett