Re: Bathroom issue, perhaps TMI
If my kid left a mess like that regularly on someone's toilet, I'd be apologetic to the person and then teach her to clean up after herself privately. The fact that mom got mad makes me think she isn't doing anything to teach her daughter and that instead she think this is somehow the secretary's fault for even bringing it up.
At 10 years old, I would to talk to the girl herself. If she hears it from her mom, she's likely to be even more embarrassed. I'd be really polite and tell her your realize it's kind of out of her control, but that she should wipe down any messes she makes.
She may not be embarrassed about the messes she's making now, but as soon as kids at school realize her problem, she'll be teased endlessly. She needs to know she can clean up after herself.