Hi there,
I would be more than happy to give you a breakdown of ticket prices.
Here's a list of ticket prices you would pay without the JR Pass.
Narita Airport - Tokyo ¥ 3,220
Tokyo - Kawagoe (return, Shinkansen) ¥ 7,220
Tokyo - Yokohama (return) ¥ 940
Tokyo - Kamakura (return) ¥ 1,840
Tokyo - Hakone/Odawara station ¥ 3,940
Odawara - Miyazu ¥ 15,760
Miyazu - Kyoto ¥ 4,630
Kyoto - Nara (return) ¥ 1,420
Kyoto - Hiroshima ¥ 11,290
Hiroshima - Kyoto ¥ 11,290
Hiroshima - Miyajima (return, including ferry) ¥ 1,380
Kyoto - Tokyo ¥ 13,800
Tokyo - Haneda Airport ¥ 650
For a total of ¥ 77,380 or ¥ 18,030 in savings, as the 21 day JR Pass is ¥ 59,350. I have not included local travel in here, within Tokyo/Osaka or travel to Osaka. So your savings would probably end up even higher.
I also assumed that you would visit Miyajima and Hiroshima in one day. If you visit both separately as day trips from Kyoyo, you would add an other 20,000 yen in savings.
All in all, the JR Pass would make for a great choice!
Hope this helps!
Daniel