Statue of Liberty and Ellis Island -- we just to the subway to Battery Park and back it was easy as pie. It is airport level security so you need to be prepared for the security line.
Dylan's Candy Bar, FAO Schwartz, Sarabeth's for brunch -- they are all within a few blocks of one another
The Metropolitan Museum of Art -- something for everyone
Broadway Show and Times Square -- Of the 3 shows you mentioned, I have seen The Lion King and I think it is for kids younger than your daughter. My friends highly recommended Wicked for my 11 year old. There has been a Harry Potter show playing the last couple times we have been there, but we have never been able to work it in.
The original Shake Shack in Madison Square Park -- the lines are long, but it is kind of cool to see where the magic began.
The 9/11 Memorial was a very emotional experience for my husband and me. I wish we had not taken our daughter.
Some place fun for dessert -- Serendipity is talked about a lot in this thread, but Momofuku Milk Bar is another fun place to get dessert and they have an upstairs lounge where you can hang out and eat.
Does she like Top Chef or Iron Chef or celebrity chefs in general? Pick one and make reservations at their restaurant.
Chelsea Piers
I know you said shopping isn't her thing, but Uniqlo is a fun place to go shop and there is not a location in DC. It isn't a girly girl shopping experience at all.