17 days in Japan

Hello.
Thank You in advance for Your help.
We arrive to Osaka Kansai Airport on September 15th (8:15 in the morning)
Then:
September 17th - from Osaka to Hiroshima
September 18th - from Hiroshima to Himeji and to Kyoto.
We will be 5 days in Kyoto during which we want to visit Koya and Nara.
September 23th - from Kyoto to Kanazawa and then to Toyama
September 24th - from Toyama to Matsumoto (Alpine Route - so JR Pass won't help on local trains and buses there).
September 25th - from KMatsumoto to Nagoya (and from Nagoya to Ise and back)
September 25th - from Nagoya to Tokyo
6 days in Tokyo when we want to visit Nikko Fuji (and 5 lakes) and Kamakura.
October 2nd - from Tokyo to Narita airport and home.

I made an approximate calculation and it seems like even 21 day pass is worth it. But my main worry, that some of the trains do not accept JR pass. Llike NOZOMI train on Tokaido line, and we have most of journeys there (Osaka-Hiroshima-Kyoto, Nagoya - Tokyo). From Kyoto to Nara and from Tokyo to Nikko JR trains are not ideals as well.

Please, advise if JR pass is recommended and which one if Yes.

Thank You again,

Igor.