Absolutely NOT. And the fact that you are considering such an idea from a 12-13 year old concerns me. I would not even entertain the idea.
A teenager NEEDS direction from their parents. Girls especially need direction from their parents to ensure that they make healthy relationship decisions, and healthy decisions about their bodies. Their PARENTS, not someone hired by a school who lives in a dorm with them. WHY would an adult live in such a situation? Because they have no children or family. I don't want someone with no kids "raising" mine!
Teens are absolutely NOT mentally equipped to be living in an apartment by themselves, going to school, AND holding down a part time job. Yes, I know it's been done....I moved out at age 17. But I really WANTED my mom to guide me. I needed her and my stepdad, I just didn't know it or want to admit it.
Moving out of the parents' house is a college level living...and even college students aren't mentally ready most of the time (which is why so many end up flunking out and living in mom and dad's basement!)
The road for kid-to-adult independence needs to be taken in small steps with lots of love and logic guidance from parents. I absolutely would NOT send my child to boarding school at age 15 when they have so much growing that needs to be done under mine and their father's guidance.
Finally, boarding schools are for kids who have issues or for kids of absentee parents who don't have time to raise their own children (or want to). I'd never send my kids to boarding school simply because it would send a very negative message to them about who they are and their relationship with my and their dad. We love them and WE want to be their parents.