I have a very shy 11 year old, but up until last year we were in very small schools (under 20 pupils) so it wasn't so much an issue as they all played together anyway. Now she is at a bigger school, although still very quiet, she is very comfortable in her own skin. She still tells me she is lonely, but acknowledges that that is mostly because she can't bring herself to move outside her comfort zone and speak to people other than the ones she knows very well.
Although it is hard for you to stand back and watch her being alone, I think it's important that you don't pressure her into being like everyone else. In reply to whatdayisit you said that you are not trying to change her but you are trying to "make her more confident". Confidence comes from within, and pushing her to speak to strange children at soft play etc is not going to give her this confidence I'm afraid!
Keep on with the structured after school stuff. Dd1 is a real tomboy and does swimming club and scouts. These structured activities give her the opportunity to be around other children without having to speak to them - sometimes just removing the need for eye-contact can work wonders (dd1 finds it easier to talk to people if they are working together on a task, for example). Could your dd go to brownies, dancing etc? FWIW dd1 is terrified of drama, again because it means drawing attention to herself; she dreads it at school.
Re. playtimes at school - could she take in a skipping rope, a tennis ball, jacks? Then if no-one is playing with her, then she has something to do rather than wander around feeling lost. And also someone might come along and ask to play with her.