Ariq, I am sure all of us can identify with your worries and anxiety about your daughter but as is being said on the posts here, it is still very early days.
There was a song in the 1950’s called ” camp granada” where a homesick boy reads out a sad pitiful homesic card to his parents as he sits inside at summer camp on a rainy day, as the sun comes out his mood changes and he ends the card ” kindly disregard this letter” . It is still September and with all the chagim etc it is hard for a young girl to be away, but it is also enriching and beneficial too. I hope soon she will settle, it is natural for kids to unburden themselves only to feel better in a short time. Be strong and try and think positive, of course a sem in NY sounds great, but to be in Eretz Yisrael is an experience second to none she may well thank you for later on!