The Statue can be seen all the way from Battery Park, but not well. I suggest you take the ferry that goes there. It also stops at Ellis Island and is a great day trip. Is also not far from Wall street area. Your niece will love FAO Schwartz. Is a lot of fun. Is not far from Columbus circle and you may enjoy a horse carriage ride in the park. In July is tricky because they don't do it in real hot weather. Of course, Times square is a must see. You may hit it on a day when Broadway is closed to traffic and looks more like a street fair.
For some shows is better to get tickets ahead, but you may look into rehearsal tickets, which are a lot cheaper. This are usually available for shows before they open. If the show is very popular it may be sold out, even by now, but at the tickets stores in Times square you can find them very cheap from cancellations.
Check this out for a few free events. they may fit your schedule:
http://www.timessquare.com/Theater/T...n_Bryant_Park/
http://www.bryantpark.org/plan-your-...mfestival.html
About Niagara Falls, Do you have a passport? Make sure you have one for everyone. The falls are seen better from the Canadian side but you need a passport to come back. Even if you see them during the day, the view is totally different at night with the light shows. The ferry ride is amazing. You can also get tickets to walk right up to the water.
Have fun.