Your sporty kids should have to learn and instrument and your artsy kid should have to do a "sport". I use that term losely.

Earthtrex has indoor climbing and classes you could take as a mom/daughter thing.
Golf
Kayaking
Mountain biking
running

Middle school is hard because sports get very competitive in this area. Does she have a friend she can do this with?

This.

http://gotrdc.org/

I think you are right to require a physical activity over the summer.

What about tennis? My DD is not into sports, but she finally took up tennis and really likes it. She does not take it with a friend and that's how she wanted it. Most of her friends are somewhat athletic and she would be embarrassed to start a sort with them. Maybe try suggesting something like that that she can do solo?

What about running? Maybe you could find a fun 5K event coming up in the fall and ask her to train and run it with you?

I'm reading this from Boston , have no idea if this is a possibility available to you: you described my daughter exactly, violin and all....and she has taken up fencing! I know! with such fervor, she is actually ranked nationally in her age group (small pond, right?) but this helps her overall fitness and, shout out to DC values, Ivies like this.