I teach social skills in a school of 750 8-12 yr olds in the us and have seen many kids with similar issues.
A few things that I have found successful with kids is: first of all, acknowlege how your daughter is feeling about school and being social (ie "you seem nervous about going to school/joining choir or whatever"), allow her to talk about how she feels and let her know that her feelings are OK..Maybe even share some things that make you feel nervous. If she's feeling anxiety (and it sure sounds like she is), a good strategy for you as the parent is to gently encourage her to participate in an activity without forcing her. So it could start wtih you letting her know she can back out at any time. If it's joining choir, maybe first say "if you want,we can sign you up, then you can decide on the night of the prectice if you want to go" , then on the practice night if she resists going, say "how about we go there and if you decide not to go in, that's ok" Keep going-- in baby steps, but always let her know, she has an out and BE SURE YOU HONOR HER WISHES so she'll know she can trust you to not force her.You can always gently nudge her though "do you think you could go in just for a little bit?, we can leave whenever you're ready". The troubling thing with anxiety is that, having someone force you to do something makes it WAY worse and also doesn't give the anxious person that good feeling that comes from making the decision to do something EVEN THOUGH it makes you feel nervous. My tag line to kids is "Be braver than you are scared" This is how I define courage. Another strategy is to play up the things she is comfortable doing, to help build her self-confidence. You said she likes to jog, can she join a track team or help younger kids learn a skill, read to younger kids at the library. Any of htese may help build her confidence which would help her feel more comfortable with her peers.
You said her brother is very social. Be sure not to compare the two as this will make her feel even worse.
I hope this info is useful.....