Finally fixed the problem! I had been using a small needle-nosed pliers to wriggle and pull the button out but couldn't get a good grip. Finally used a hemostat I had laying around which was much grippier. After several attempts over several days the button moved a little, and with one good yank the thing finally came out. Works OK now, but still can't figure out how it happened in the first place. But at least I know how to fix it now.
Thank you EDBwoy for your support.