My heart goes out to you because, as a mom of a ten-year-old daughter who has sensory issues and finds it very hard to make friends, I can SO relate. My daughter has been in dance for seven years but feels like an outsider because most of the girls go to school together (she dances in Monona but goes to school in Oregon). However, she loves dance so much that she's decided not to let it bother her. She also just switched over from ballet to jazz and has discovered that she loves dance even more now.
This year she started band (5th grade--tenor sax) and, while it has been shaky, I do believe she likes it. I signed her up for two special 6-8 wk courses that they do in school (probably now in fall/winter), stamping and yoga, which they'll do during the lunch hour. She loves horses and goes to visit and ride at a friend's place who raises/supplies the horses that my husband uses for his Civil War Reenactment hobby. She is an only child and we live in a country subdivision with not a lot of kids her age, and our families are distant, so... well, you do what you can do.
During summer break I keep her busy with a week of Vacation Bible School and then four weeks of summer school classes through her school district. In August, when she had nothing planned, she worked on stamping and counted-cross stitch projects, which she'd learned to do in summer school. We also took a few one-day trips around the state.
One of the problems I've consistently run into is the fact that it is so hard--sometimes almost impossible--to get playdates set up for her. Everyone is so busy with all of their own activities--and then their parents are busy with activities or work--that even just trying to get her together with the few friends she does have is nigh on impossible.
She's very close to me and her dad, and right now, we're trying to help her develop a more positive attitude (she can be quite negative and down on herself). She tends to feel insecure if a friend of hers makes a new friend, because then she feels their "friendship" level is threatened. However, she just told me that she's getting to know the new friend her best friend has made, so things are looking a little more positive.
When she was younger I cultivated friendships with other moms/kids close to her age around town/in other school districts, so that she'd have friends outside of her school friends. That worked for a while, but now they have all gone on to make other friends in their school districts and the friendships have sort of faded away. Technically, they didn't have enough interests in common to hold the friendships together.
As she gets older, I keep trying to find things that I think she'd like to do with me. But it is hard, because I can't make friends for her. I didn't have a lot of friends when I was in school either--most of my friends were older or younger than me. I didn't have many friends in my own age group. But I also lived in town, which I think makes it easier to make friends than living in the country.
The best advice I can give you is: Be there for her. Let her know that you love her, that she is a very special and unique person. That school only lasts until she's 18, and then she has the rest of her life to find out her passions and to do what she wants to do and make the kinds of friends she wants to make (as in, she's not limited to who is in her class/school).
If you can find anything you can get her involved with that is a community project--even if you do it with her--that would have such an impact on her. My daughter has asked me if we can find some community thing we can do together, so we're looking into that idea (just so hard to do, what with school and homework and dance lessons and sax practice and...there's only so much time in the week).
Do everything you can to cultivate her interests. But also keep in mind who your daughter is, whether she is an introvert or an extrovert. You don't want to push her more than she's ready for.
I still remember the girl in my class who was the quiet, shy, wallflower type. Then we got to see her Senior Class picture--and it was like she was a totally different person. And she was. Who she was in school was NOT the person she really was. She kept to herself and skated by--I can't for the life of me even tell you if she had any friends at all--but it was very obvious that she had a very rich and full-filling life--probably even had friends--outside of school. I've heard she's gone on to have a very successful career and doesn't lack for friends.
I always remember what someone told me: a person really only has a few very close friends. What a person has more of are acquaintances.
If you'd like to chat more, contact me. Maybe we can get together and see if our daughters click/have interests in common. I can always use new friends as well. ;)