Which pass should I buy?

14 day pass pays off and is suitable to that itinerary

To make 7 day pass suffice, you need to rearrange your itinerary

(7 day JR Pass) Tokyo - Kyoto
Kyoto
Kyoto
Nara Day trip
Kyoto to Mt Koya
Mt Koya - Himeji - Hiroshima
Hiroshima/Miyajima. evening train to Tokyo(or Hakone)

(3 Day JR Kanto area pass) Hakone to Nikko
Nikko to Tokyo
1 day side trip to somewhere(Tomioka Silk Mill or Fuji five lakes?)

A 14 day JR Pass would pay off. Around Hakone though there is little you can do with the pass. Look into a Hakone Free Pass.

http://www.japan-guide.com/e/e5200.html

http://www.japan-guide.com/e/e2358_008.html

For your Days 15-16, there are many possibilities. Just beyond Miyajima is Iwakuni, with its iconic bridge and mountaintop castle.

https://www.youtube.com/watch?v=0COOZXi66mM

You could even go farther into Yamaguchi - the Akiyoshido Cave is a 1km 300 million year old limestone cave that is just stunning:

http://www.japan-guide.com/e/e6130.html

If you'd really like a nice half day hike, Kotohira in Kagawa offers a beautiful mountaintop shrine and view, but it's the trek up that is the real adventure:

https://www.youtube.com/watch?v=ley9rVIewoQ

Okayama is between Himeji and Hiroshima, and has one of Japan's Top 3 traditional gardens, as well as the nearly unknown Handayama Garden:

https://www.youtube.com/watch?v=5zAI3_N7NEM

https://www.youtube.com/watch?v=JGQ8o7k9GbA

Onomichi in Hiroshima is a rather off the beaten path city with a hillside temple tour, as well as the stunning Kosanji Temple which is like a Buddhist Disneyland:

https://www.youtube.com/watch?v=97Q5EjEOXI8

You could also take a trip going back by way of Kanazawa, which is a beautiful city, and take the recently completed bullet train line back to Tokyo:

http://www.japan-guide.com/e/e2167.html

Hi there,

Fk and Toraneko have covered most options.

To add my own suggestion. Consider going up to Matsue from Hiroshima. It's very nice place to visit, the Izomo Taisha Shrine and Adachi Museam of art are not to be missed!