I did the same as you many years ago, it wasn't a cat but polished stairs and slippery shoes. sadly it did take AGES to heal which isn't what you want to hear. What helped me was arranging cushions with a gap at the back where my tailbone was when I was sitting...you might buy a piece of upholstery foam and cut out a U shape? and using ibuprofen gel as often as allowed on the area where the pain was....If you can then sleeping on your side or your stomach can help but its hell when you fall asleep and roll over. I guess it took around a month for the pain to settle and about six months before it went completely....sorry. more info here, pretty much confirming my experience. They mention steroid injections in the text, but when I looked into this there didn't seem to be much good research evidence that they helped....I dunno, maybe the treatment has improved.
https://www.nhs.uk/conditions/tailbone-pain-coccydynia/