I was just like her - I hated team sports and wasn't confident or coordinated enough to shine at individual sports. I was a book worm too
I discovered running... it wasn't until high school, mind you - I don't know if I would have done it at your daughter's age. I love it though! It's solo (ideal for the sensitive introvert), requires zero coordination (pretty much, anyway), and you don't have to do it competitively to reap the fitness benefits. You can do it anywhere - at the school track, at the park, through city streets, or in your own back yard if it's large enough for laps.
Is there any way the two of you could run together? If you're not in running shape you'd have to start slow, but it's totally doable. Or if you have a yard big enough or a safe place she could be by herself she could run to music with an mp3 player - I love this - it's a great stress reliever.
Good luck (& have fun :-)
