Whether you purchase a 7 or 14 day rail pass very much depends on the amount of JR travel you intend to do. To make a 7 day rail pass worthwhile you need to spend over ¥28,300 on JR tickets. For a 14 day rail pass, you need to spend over ¥45,100.
Looking at your itinerary you're going to be in Japan for 12-13 days, which will need careful planning. Structuring the trip for this amount of time I would try and base the travel around a 7 day rail pass. You'll have lots to do! For example:
- Tokyo (Using [NEX+Suica deal], or Tokyo Metro day passes + Skyliner deal)
- Tokyo > Hakone (using Hakone Free Pass)
- Hakone > Tokyo (using Hakone Free Pass)
- Tokyo > Takayama (First day of 7 day JR Pass)
- Takayama > Kyoto (early train)
- Kyoto (possible day trip to Nara?)
- Kyoto > Miyajima (island off Hiroshima, leaving big luggage in a coin locker in Hiroshima station)
- HIroshima > Tokyo for flight out.
If you have other plans, for example, to visit Nikko, or if you really aren't that interested on staying in and around Tokyo, then a 14 day pass would be a good option, but the above itinerary is well spaced and will be a nicely rounded trip.
Hope that helps!