With regard to entertaining a fourteen year old daughter, most of the activities I can think of are water based entailing boats so the seasick thing is always there.

I'm not sure if she'd enjoy the zip lines but most kids do as well as an early morning trip to the Butterfly Farm, and then there's shopping at the Marketplace in Marigot or in P'berg. Airplane watching in the afternoons at SSB&G usually appeals to kids of all ages but please don't let her hang off the fence there. And then there's the vast variety of beaches, so many of them with their individual entertainments - we always found Orient to have the most between the trampolines, restaurants, jet skis and parasailing. Unfortunately, I'm not very up to date so not sure if those options still exist there.

If you want to take her to another island, you might consider Pinel over AXA.