|First paragraph: I was focused on the murder and had to have "the investigator of the murder" thrust upon me while still focusing on the murder. Suggest a better transition from one (the murder) to the investigator. Also, I'm fairly certain an individual in a deerstalker cap isn't what most people picture when thinking of a murder investigator.
2nd paragraph: Continuation of forcing a particular image of the investigator and transitioning to the school girl sleuth. Holmes (or a knock of of him) is unique to a specific type of reader/genre whereas just a policeman is the default for most. The only time a private investigator would get involved is at the behest of a patron or request of the legitimate authorities.
3rd paragraph: Are we supposed to take the statement she's forgotten her given name seriously? I get what you're going for but that's a stretch, even in a story. Also, I was confused a bit because she refers to herself as a farm woman just after we've been informed she's a school girl. Not a "big" deal, but the inconsistency is distracting. Also - a school for polite "and" petite girls? Does such a thing exist?
4th paragraph: So, is she British living in America or native born? She's in the eighth grade, right? Have her internal dialogue read/sound as such. Is this how she speaks in her every day conversations? I ask because, as a born southerner I'm pretty sure this wouldn't be how she would likely speak - even to herself.
5th paragraph: Pronouns. After the second "Ms. Jakob" I was mentally inserting "she".
6th paragraph: Ms. Jakob refers to Rye causing trouble but all she was doing was looking out the window. Not to nitpick but once again, it's distracting not having the action/events match the accusation/reference.
7th paragraph: Well. A bit of an over reaction to being call Mrs vs Ms but, okay.
8th paragraph: Glacier Town? A town called that in the south? by the way - what state is she in? "In the south" seems unrealistically unspecific except if you still haven't decided which one you want it to be. Once again, the lack of specificity for something so routine is distracting. Where is she? Tennessee? North/South Carolina? Virginia? Also - mess hall is specifically a military term. I think cafeteria or lunch room is what you are looking for here.
9th paragraph: She was "sent" to a boarding school in the U.S. suggests she's from somewhere outside the country but you say "I've lived in Glacier town all my life and even I was freezing." Which is it? Was she sent there or has she lived there all her life? I ask because, it's going to affect the details of your narrative this far and going forward. The inconsistency is also (you guessed it) distracting. Also, why is she waiting on her?
10th paragraph: Judgmental grimace. Still wrapping my mind around that one. She's "judgmental" but being tentative? Usually, someone being "judgmental" is coming from a position of superiority. Seems shoe horned in.
I can see the tone you're going for and it's obvious you've been heavily influenced by a few different genres (mystery, YA fantasy/possibly horror) because the story thus far is chock full of the tropes and character descriptive. Not sure if this was a first draft or not but, need to go back and check for inconsistencies and formatting. You also need to decide what this is going to be; where the protagonists is from; and what her narrative voice is going to be. Is she an adolescent teen or a middle aged/older British woman (ala Mrs. Marple) stuck in the body of a southern teen who is possibly a transplant from another part of the country?