Hi,
You have 2 choices for your time in Tokyo. If you want to take more than 3 day trips and the total regular fare would cost more than the difference between a 2 and 3 week pass (about 13000 yen), then getting a 3 week JR Pass would make sense. But if only 3 or fewer day trips, then get the 2 week JR Pass and for your time around Tokyo get a JR Kanto Area Pass. There are many areas near Tokyo for you to consider - there is Kamakura, NIkko, Kusatsu, Izu, Karuizawa, the Boso Peninsula, and many others. For Hakone, look into getting a Hakone Free Pass.
Because you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5 PM, you can zoom over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are magnificent. For your days, there are many great places, including Kobe which you mentioned as well as Himeji (with Japan's best castle), plus an overnight at Koyasan would be an unforgettable experience. In addition, Onomichi is famous for its temple tour and wildly ornate Kosanji Temple. Nearby Kurashiki is a famous own city for its canals, and Okayama also has its own castle and one of Japan's other Top 3 gardens, Korakuen.
For Hiroshima, Miyajima is by far one of the biggest tourist draws in the area, but aside from the Itsukushima Shrine with the giant "floating" torii, the Daishoin Temple is also a superb place you should not miss. There are several other pagodas and temples, and climbing Mt. Misen offers a fantastic view over the island and whole area.
Nara is also a must-see place, although there is far more to enjoy in the Kansai area that you can safely skip Nagoya which has comparatively very little.
Best of luck.