I did the Rockettes with friends...it was very meh. Not my thing at all, but might be something you and she would like to do. Runs about 90 minutes. On Broadway, I recommend Six (it is only 90 minutes) and &Juliet. Empowered female characters and great music. Great for a girl her age. Spamalot will also be doing a revival at that time, but I think you would enjoy that more than her. Definitely get tickets to Six. Maybe see what else you want to do before tying up your time with tons of shows, but if you want to fit in a second, &Juliet is the right answer.
I am not an ice skater, but dealing with the crowds at Rockefeller is a beating (I walk through there to get to my office). Book as soon as you can. Also, you can also skate at Bryant Park (Definitely get hot chocolate at Angelina while at Bryant Park and browse at the winter village). If Christmas Markets are your thing, the Union Square one is massive, but it and Bryant Park have similar vendors.
Definitely see the windows. Bergdorf, IMO, has the best. But Bloomingdale's and Saks are also relatively close. You can make it a walk. If it snows before Christmas (which it has once since I have lived here), then Central Park is a great place to go hang out.
If Art is something you guys are into, the Met is always a good idea. https://www.metmuseum.org/exhibitions/upcoming Some good exhibits will be on in December.
Probably tons more going on in December, but it is way too early to know about a lot of them.
I am actually in London right before Christmas onboarding a new employee (and I am going to try and hit some Christmas markets the weekend before..thinking Cologne and Aachen).