I guess most of us are hesitant about offering advice that might result in you making more damage.
This is the first time that I have seen under the button.
If it were me (note "ME" I'm not suggesting that you try it) I would mark the plate orientation and partially unscrew the plate so that I could determine if the brass part was coming away with the plate. If it did then I would carefully remove the plate and hope that the underside of the brass part was hollow so that you can poke the broken part out.
If however the brass part appeared to be staying where it was I would:
a) keep asking for help hoping that someone has already had this problem and fixed it.