Like Jersey Girl said, the bald patches, irritated-looking skin is urine scald. It has developed because that area of skin has been in frequent contact with urine.
My bunny Moshu developed a little bit of urine scald because of UTI. I apply her some Neosporin ointment twice a day (minus the pain relieving stuff ; should only contain Polymyxin B sulfate – neomycin sulfate – bacitracin zinc ). You can also apply Calendula cream/lotion or Silver sulfadiazine.
However, application of these ointments will help much more when that scald region is dry. It wont help too much if you are applying the ointment but the scald is still exposed to frequent wetness.
I posted a similar topic a few days ago : https://binkybunny.com/FORUM/tabid/54/aft/133991/Default.aspx
Since Bugs is very well litter trained and he’s peeing all over all of a sudden, it could be urinary incontinence. He may be having difficulties controlling his bladder and hence, the leakage. Moshu used to experience that two months back. Every time she would hop, she would spill some pee. As Eepster said, its definitely vet trip worthy.
Urinary incontinence can be a resultant of UTI, E.Cunniculi and perhaps a few more conditions. If its UTI, the vet will prescribe him some antibiotic (mine is on Enrofloxacin, some are given Ciprofloxacin), if it is E.Cuniculi, they will give him Fenbendazole.
Only when what is causing the urinary incontinence is targeted and combated will the situation solve, your constant efforts to clean the cage will not help too much in the long run and will only drain you of your energy. Till you find a good vet who makes the right diagnosis, you will have to continue sweeping and drying her cage often. You can start applying either of the three ointments too.