I’m very lucky in this regard. I have a second daughter, she’s proper grown up and lives her own life - but she still calls me just to tell me about her day … it’s amazing. I never would have dreamed to do that with my parents, our relationships are so different… I mean, my younger one is a teen, so she rolls her eyes at me at least three times a day, but that’s her job, right. 😄 All in all, we get along great. I mostly try to stay out of her hair, but still be curious about her - it’s a balancing act for sure.
Children love to help. It comes to them as naturally as play. “Chores” are just things that need to be done … I think it’s helpful to not think about something you have to do everyday as a burden, just for my own mental well being, haha. When my daughter was young and around me all the time, I would have found it weird to exclude her from everyday things that just need to be done. I didn’t even phrase it as “help” - we would just do those things together, just like we played together. She’s a teen now, and knows how a household works and she just does stuff that need doing, just like any other member of our household. I mean, of course her room is a mess, but that’s just age appropriate.
Children are very good at adapting to new things. I think the most important thing is to keep the lines of communication open - maybe you already have a ritual together where you decompress together and reflect the day, maybe expand that or introduce it, so she has a set “room” for her thoughts. More than “doing something”, I would encourage you to listen and see what she needs (not that I think you’re not already doing that, but sometimes it can feel like we have to ACT, when being attentive and react can be more helpful).
I personally would try to get involved in the school, as far as this is possible, get to know the other parents, get a feeling for how the community works, where/how we can socialise and hopefully get some ideas how to help her connect to the other children quickly and easy (events, clubs etc). Maybe the teachers already have some ideas on how to help integration - when a child from another country joined the school of my daughter, they did like a week or so of days where the children would show and tell something about their country or culture or family, all of them, not just the new one, so they immediatly found things they had in commonm and others to be curious about.
Be open, be curious - she’ll pick up on your attitude towards the new situation, so make it a fun adventure more than a challenge. :)