For the brainy, artsy, 13-year-old, I highly recommend Camp Arena Stage, which is a general arts camp, not just a theater camp. It's very well-run, with wonderful counselors, and there are two sessions--one four-week, one two-week--to choose from. My now -14-year-old daughter has gone for five summers, and truly loves it.

I'd try Glen Echo:
http://www.glenechopark.org/summer-camps

Pretty convenient locationwise for DC, MD or VA but not sure where you are.

Maybe Casa at the Sheridan School: http://www.casaatsheridan.org/casa.html

I am s/o day camps for a sporty, musical, academic 10 yo boy and a day camps or away camps for a brainy, artsy, bilingual 13 yo girl. Any recommendations based on experience?