Lion King and Mary Poppins come to mind - esp for the 10 year old. The 13 year old might find the latter kind of childish -if she's relatively sophisticated.

Lion King
Mary Poppins
Peter and the Starcatcher
Wicked

Your girl would love Wicked, but there are likely no tickets left (at least good ones for regular price). A lot of kids like Stomp, some like Warhorse (though it may be too intense for a 10 year-old, though it's based on a kids book).Kids like Spiderman, and it's often available for a discount, but it could be too intense for a 10 year-old.

We just got back from NYC on Sept 3. Took our two girls, age 18 and 10 to Mary Poppins and they both loved it. Got tickets at the box office on day of show for $85.00 each for fairly close to the front of the mezzanine. We could see everything well and Mary flew right by us with her umbrella. There were tickets for $62 each at the back of the mezzanine as well but we felt it was worth the extra $20 each to be where we were. I think there was enough action that your boy would like it too.

I didn't like "Peter & the Starcatcher" -- and I think kids would hate it. Although it's based on Peter Pan, the show isn't geared for kids.

Go for "Lion King" for sure.

At that price, the tickets that oregonmom mentioned were probably available through a discount offer from either broadwaybox.com or playbill.com. Thus, I would recommend checking out those websites. I seriously doubt that you could just show up at the box office without a discount offer in your hands and get tickets at that price.
Mary Poppins would be a good choice, as would THe Lion King, though there are no discount tickets available for the latter.
I defihitely would NOT recommend Peter and the Starcatcher for the young ones.

That's interesting on the opinions on Peter and the Starcatcher since my preteen grandkids loved it!

It really depends on your kids. I wouldn't get stuck on hunting down a "kids" musical unless you're sure that's all your kids could handle. We went to Nice Work if You Can Get It. It's an old-fashioned musical with big dance scenes, light plot, but that also meant it had a lot of action. Even better, it also had a lot of slapstick humor and my 11-year-old thought it was really funny. We did Ellis Island/SOL trip during the day and the musical that night. He told me that was his favorite day of our two-week trip. And I'm sure there's several others he would have enjoyed but we didn't have the time or the budget. My daughter, 14, and son, 16 also loved it but I knew they would.

We had planned to go to TKTS to get our tickets but I had mixed up the opening time of the booth in Times Square with the one at South Street Seaport. Since the theater was only a couple blocks away, we walked there and bought them directly from the theater for that night. Only had to wait while one couple bought theirs before us. Spent the rest of the day at the Met and went to Mary that night. I did not have a discount coupon or anything. That was their price.