D.B.
I agree that deeply absorbed pee is not coming out with surface cleaning.
I don't know a lot about dog pee (except that my dog deposits it in some pretty inconvenient locations!) but maybe it would help to know that human pee is STERILE! I learned that on Dr. Oz - in fact, he said military personnel are taught to clean wounds in the battlefield by peeing on them when no water is available! There's definitely an odor, but barring infection, the urine is sterile.
Might make you feel less grossed out.