As you already know, being checked out medically is always the first step. Jack, who incidentally looks like Jasper, once had a UTI and started urinating on my bed, even when people were still in it!!! We did a urinalysis and found out the poor dear had the UTI, got him on antibiotics and the peeing stopped.

If everything checks out medically I'd highly recommend reading this website....
http://www.catinfo.org/litterbox.htm


It's a long read but just about covers everything litter box related. In addition I highly recommend a rubbermaid tub, Cat Attract litter, Feliway spray, and a good enzyme cleaner for the already stained areas.

Good luck.