Hi there,
To see if a JR-Pass would be suited, here is a little breakdown of ticket prices
Kansai Airport --> Kyoto ¥ 3,490
Kyoto Nara (return) ¥ 1,380
Kyoto --> Hiroshima ¥ 10,790
Hiroshima Miyajima ¥ 940
Hiroshima --> Odawara (for Hakone) ¥ 17,530
Odawara --> Tokyo ¥ 3,640
Tokyo --> Narita Airport ¥ 2,940
14-Day Pass option:
For a total of ¥ 40,710. The 14-Day JR-Pass is ¥ 45,100. It would be worth getting the pass if you could do an additional ¥ 4,390 worth of travel. For instance with a day trip to Nikko or Izu would be perfect for this and you can also use local JR-Lines in Tokyo.
The way to Hakone is covered by the JR-Pass but does not include travel in the area itself. For this a Hakone Free Pass is recommended.
7-Day Pass option:
This is also a good option and may turn out to be a bit more economic,
You could fit your major travel in the 7-day period and buy local tickets/passes for the rest.
The best way to use the 7-day pass would be to use it for the following legs:
Kyoto Nara (return) ¥ 1,380
Kyoto --> Hiroshima ¥ 10,790
Hiroshima Miyajima ¥ 940
Hiroshima --> Odawara (for Hakone) ¥ 17,530
For a total of ¥ 30,640 saving you at-least ¥ 2,340 over normal tickets as the 7-day JR-Pass is 28,300.
Day trips: Besides this you would then use a Hakone pass to Tokyo and single tickets for the rest.
When you arrive at Kansai Airport you may also find the Icoca + Haruka ticket combination useful.
All the options you mentioned are great!
If you would like to see more of Japanese History and tradition go to Nikko, the temple complex build there is one of the most impressive in Japan. It makes for a very good day trip and you could use the 2nd day for an other destination like Kamakura or simply enjoy Tokyo a little more.
For Onsen and beautiful coast lines go to Izu, the area can be stunningly beautiful and is a great place to forget about the rest of the world before returning home.
Personally I would take the 14-day pass, this because being able to take any train anywhere gives a really nice feeling of freedom and it takes the worries away of tickets and budgeting trips. Lastly, if you are unsure about your last couple of days, you can just decide them on the spot and do what you feel like.
Hope this helps,
Daniel
