decrepit said...
Don't like your chances, drilling into a broken stainless screw isn't easy for a start.
You're spot on there, you can however drill a hole in a flat sheet of steel, and press firmly over the screw, using the flat as a template, then simply 'tap' a precision flathead screwdriver, slightly wider than the screw's diameter (oldie/cheapie, incase it breaks

), this forces the hole to become elongated, and simply unscrew...
...this is incase you don't have a dremel