Make sure she doesn’t have a UTI, make sure she has a space safe from your dog. Make sure she has at least two litterboxes, one on each floor of the house. Use cat attract litter. And make sure those boxes are spotless.
Often, inappropriate urination is due to a UTI and can continue afterward as they associate pain with the litterbox. If the above tricks do nothing to curb this, and you’re 100% certain she’s clear of a UTI, you may want to consider changing the box type as well as litter type. Dr. M has mentioned in the past that plain old dirt (like from the garden – pesticide and fertilizer free, of course) can attract cats.
An update: he’s calling two local rescues for help. I’m leaving this up, however, so folks can share input should someone come for the same situation.