Osaka is probably more in line with your priorities. Kyoto is more about traditional and historic sites while Osaka is very modern and vibrant. The two cities are only 1/2 hour or so apart by express train so a day trip to Kyoto should take care of your interest in a castle (Nijo Castle) and a few temples and shrines (literally hundreds to choose from.) For an even better castle experience, you could take half of one of your days and go to Himeji or Hikone. Himeji is the big, impressive and famous castle but I personally like Hikone better. It is smaller but it sits on the shore of Lake Biwa and has some beautiful views. The little town of Hikone is charming and less crowded with tourists than Himeji.

You might also consider a half-day trip to Nara, which was the ancient capitol before Kyoto, over 1200 years ago. There is a lot of old and traditional Japan to see in Nara but the large park surrounding the Giant Buddha temple (Todai-ji) is pretty and has tame deer begging for hand-outs that might amuse your child.

For your itinerary, the 7-day Japan Rail Pass would likely be a good investment. If your daughter will still be 11 at the time you purchase the exchange order, she will qualify for a child's Rail Pass, which is about half the price of an adult Pass, saving you some money. If you activate the Rail Pass for the 7 days before you fly home, counting departure day, it will cover local Japan Rail travel in Tokyo during that period, like the Yamanote Loop Line and any day trips outside of Tokyo, such as to Nikko or Kamakura. After Tokyo, the Rail Pass will cover the round trip Tokyo-Osaka-Tokyo on the bullet train, day trips to and from Kyoto on the bullet train, other day trips from Osaka, such as Himeji, Nara or Hikone, and the trip out to NRT for departure for home. That last is not so important if you will be flying from HND but can save several dollars to NRT.

http://www.japanrailpass.net/en/

While in Tokyo, you should plan some time Sunday afternoon in Harajuku so your daughter can experience the youth fashion that is so famous.