This website has some great suggestions for Tokyo, especially if you're a bit adventurous and want to experience things really unique to Japan:
Tokyo travel guide for essential sights & attractions in Tokyo Japan - UnmissableTOKYO.com
If been to most of those places myself and can thoroughly recomend them.
I've not been to Miyajima, but have heard that it's a bit of an overrated tourist trap.
I liked Nagoya - it has a really nice castle, and a nice more relaxed feel to it than most Japanese cities - but other than that there's not a whole lot to see in the way of tourist attractions.
I definetly would keep Hakone - hot springs are one of the best and most unique features of Japan - and Hakone has heaps of them - plus the national park is quite nice as well.