Cats are disgusting little mammals. I wrote a paper in college for my advanced pathology class called "748 Reasons To Hate Cats". That listed the number of microbiological lifeforms that they carry that are pathological to human beings.
Start by buying a litterbox that automatically disposes of the cat leavings. Then start aversion training the dogs away from the cat's litter box. This can simply be done by making an annoying sound and gently pushing the dog away when it gets near it. The dog sees it as your exclusive territory and will leave it alone since you are the alpha.
I adore my cats and dogs, but I made the boundaries very clear when each of them entered the house. Make sure that the cat's box remains clean or it will find your laundry basket in which to do its business. You will NEVER get the smell out.