My 13-year-old son struggles too. He is kind, caring & clever. I have done all sorts of things like yourself engaging other families, hosting, taking kids elsewhere it is just rarely reciprocated. I had an incident several years ago, we had often taken this child to local pool that we had a membership too. It seems his family had finally gotten their own membership & he was there with school friends my son went to hang out with them & the boy said I only play with him because my parents make me! I can't say anything to these parents because I've tried in the past they laugh it off & the father lacks social graces as well; the apple doesn't fall far from the tree!
So I send my kid to overnight camp for part of the summer so we can meet other kids & take advantage of activities and experiences that he wouldn't be able to otherwise. This may sound bad to say but I decided that I would prefer that my son stay home and play on his iPad rather than play with some of these kids that really do not play nicely. It's so bad that even my older son commented how mean they are to each other!
And let's face it now that were adults, do we hang out with friends daily? If we work with others aren't there days we come home and feel like we just don't want to deal with anybody? Does that make us loners? No?
Have her walk/bike around the neighborhood work on her interpersonal skills let her talk at the neighbors maybe she'll chat with an older kid or a younger kid. Take her to the store with you. She can find other ways to be happy. Our kids see other kids hanging out with each other and they think that's what they want. But in reality, it can often becomes a catfight for who gets control of what they get to do.
So the host and occasional supervised hang out and invite a couple girls over for a few hours but don't feel like you have to do it every day and put your daughter in an activity that she enjoys. I'm not saying it's easy it's still hard some days.
Yesterday I agreed to let some of these kids come over last minute when they weren't allowed at this other kids house. It was almost 6 o'clock I was starved wasn't prepared to feed five other kids dinner. They were in my basement as it started raining they were still so loud (my older son often has groups of kids over and it is rarely that loud!). I had to call up one kid who I've known for years to talk to him about his language which sound like it came right off the boat from deadliest catch. When they left even my son was glad!