[Electronic M]

Yikes. I wonder if a jumbo one of those stripped screw extractors would remove it...those usually have you drill a hole in the fastener then the tool has heavily tapered reverse threads so if you spin it counterclockwise in a drill it grabs the fastener and unscrews it....if you got one of those tools where the tip of the taper went in the hole in the broke off thread but the back didn't it'd probably get the thread of the filter out.
BTW the carb on my Lincoln uses that same type of filter... It's mesh is so appallingly coarse (literally metal window screen) I installed one of those clear plastic ones the auto parts stores carry infront of it.... incase anyone ever dumps coarse sand into my tank.