Gloves are a necessity when doing any work with any bodily fluid.
Sounds to me like the cleaners wouldn't stand a chance in an elementary school.
You know... vomit here, urine there, boogers elsewhere...
If you're able to, get the manager and the cleaners in a meeting. I mean... who in their right mind enjoys human waste on walls? If the management says, "I'll handle it," you need to make sure they do. People have that little tendency to say they'll do something and then blow it off for the longest time.
Eat. Skate. Sleep. Repeat.