Stuck screws are about as much a pissoff as a small part falling down a rabbit hole, or hiding a part or tool from yourself when you know the thing is within arms reach.
Try soaking the screw with some Seafoam Deep Creep and let it sit for a couple of days. Hopefully you can dig out the extractor and try another one.
When I was a kid I had a Yamaha GTMX80 (72cc minibike) that had a bolt break off in the transmission. Then of course the extractor broke off in the bolt. It took me two years of on-and-off drilling to get the piece out. Luckily you can just hack off the whole turn signal assembly and start over if worse comes to worst.