When mine broke off. I took it to the local gunsmith. He put in a stronger screw and put lock tite on the screw and thimble. It has been just fine ever since.
It's undoubably broken off to where you can't get a hold of it? I would say an EZ-out but the screw is really small. If you don't trust yourself with a punch and hammer to try and unscrew it, I would take to a gunsmith.