Hi Housewife, Welcome!
I have repaired and restored several old porcelain floors, such as yours. We used professional restoration techniques which involved using diamond abrasives. You were on the right track, when you mentioned using sandpaper.
I would recommend using TYW cleaning products 1st to remove all surface staining. Check the results after dry. If that isn't satisfactory enough, you'll need to get more aggresive. This may require the aide of a professional.