I looked at some wall mounted units last year for a very moist basement.

You'll need to spray the cleaners and soak everything so it can get into crack and crevices where the urine may be and keep them moist if possible because the enzymes are destroyed by drying out.

The inspector said if there was more ventilation in the roof, the issue "should" be solved, because it appears the moist air is just hanging out in the attic.   

Her cats peed on all the carpet in the house, it was especially moist in the corners of the living room.

Like the foundation and structure would get damage easier as the salty moist attack?

Unfortunately, I can see where the wood is rotten (from the carpet tacks) and where it was moist but even after tearing apart the walls I was unable to locate where the water is coming from.

You'd think in the scheme of things this would be an easy thing to solve, but I've tried in vain to come up with a good key labeling system and I'm hoping someone has got a solution.I've tried frog tape on the key heads, which sticks great, but even with pigmented ink, the writing wears off after a while and becomes illegible.I've tried those circular stickers that people use for garage sales. 

We have a child with asthma, so we can't have moist basements in our house.

Also, in NY inspectors are no longer allowed to test for mold so do your due diligence in identifying any moist areas or potential places mold can grow at high levels (inspectors will normally point out any suspect spots they just can't run a mold "test" as they are not certified to do so).I normally use inspectors based off referrals (maybe you have a fellow home buyer who can recommend one to you).

It doesn't matter how well ventilated your attic is - warm moist air from a bathroom hits cold air in the attic = condensation.