Hi Andy,
I have made a quick overview of your itinerary.
Narita --> Tokyo ¥ 2,740
http://tinyurl.com/c3kftqk
Tokyo - Osaka ¥ 14,050
http://tinyurl.com/cs8r5pk
Osaka - Hiroshima ¥ 10,150
http://tinyurl.com/brneb43
Hiroshima - Hakone ¥ 17,330(+300 Yen for private railway)
http://tinyurl.com/cul7u5k
Hakone - Narita about ¥ 5,000 yen depending on the route.
http://tinyurl.com/c59uxar
This adds up to about 49270 Yen, a 14 Day JR pass costs 45,100 Yen, so you would already save 4170 Yen. Please note that I have left your local travel in Tokyo and the Kansai area out as I don't know your exact plans there but this could add even more to your savings!
You could also decide to take a 7-day pass instead to cover your major travelling (Tokyo - Kansai - Hakone). A 7 day pass costs 28,300 Yen and normal ticket during this period would cost you 41.530 Yen so you would save 13230 Yen and your local travel in the Kansai area would also be covered. In this case however you would have to leave Tokyo one day later or go to Hakone one day earlier as the pass only covers 7 days of use.
Since your total itinerary is more than either pass covers, I would suggest on the first day of your arrival not to use the pass and opt for the Suica & Narita express ticket. This is a pretty good deal as it will get you to the heart of Tokyo and give you a pass with 1.500Yen credit that you can use to travel around in Tokyo. Also the Suica card is a great souvenir!
Either option would save you money, if you are looking for he most economical approach I think the 7 day pass option would save you slightly more but the 14 day would be a lot more convenient and you would not have to change your itinerary.
I know its a lot of information, so if you need more clarification let me know!
-Daniel