3 weeks trip to Japan

Hi,

I'll be travelling to Japan for 3 weeks and I'm not sure which JR Pass suits me the best. I'll be travelling from the 3rd of September until 22nd of September.

My plan is:
3rd arrive at Narita and stay in Tokyo 3rd-8th. During this time I plan to do at least 1 day trip( to Nikko and hopefully few other places north of Tokyo).

8th-13th I'll go to Kamakura and do 1days trip from there (Yokohama city, Hakone, Mt. Fuji).

13th-19th Kyoto. During this time I'll be visiting Nara, Osaka and Kobe (with possibility of staying overnight in any of these locations) and 1 night in Hiroshima - Miyajima.

If the time will allow it, I will like to go to Fukuoka 19th-21st, if not I'll be going back to Tokyo and spend the last 3 days just enjoying the city (I'll leave early on the 22nd).

I was thinking to buy a 14 days JR Pass and used it 6th-19th to cover for the cost of most of my trips. Most likely that I'll have to buy a Suica + NEX return ticket (about 5500¥).

Please let me know which one will work out better: a 7 days JR Pass or a 14day JR Pass. I mean, I kind of know that the 14 days will be better, it's just that I was reading a lot of comments and I'm not sure if all the locations (especially Kamakura, Hakone, Nara) I plan to visit are fully covered by the ordinary JR Pass and without having to pay any extra seat fee. Or if it's better to buy normal tickets for part of my trip instead.