JR Pass recommendation

We are visiting Japan for the first time at the end of April (2 adults, 1 child - 4 yrs old, need a child pass as he is tall for his age and can't sit on lap) for 15 days and I would like to seek your help and advise whether our itinerary is worth the 14-day JR Pass or if the 7-day would suffice (will start to use for Kyoto, Osaka and Hiroshima/Miyajima travel). The rest of our time will be spent around Tokyo. Here is our planned itinerary:

Day1: Arrival at HND, check in to Shinjuku hotel
Day2: Hakone, Mt. Fuji day tour (is JR Pass valid?)
Day3: Tokyo all day
Day4: Shinkansen to Kyoto (Fujiya Ryokan)
Day5: Kyoto all day
Day6: Kyoto all day
Day7: Travel to Osaka (Osaka Airbnb near Shinsaibashi)
Day8: Osaka all day
Day9: Hiroshima/Miyajima day trip, travel back to Osaka
Day 10: Shinkansen to Tokyo, checkin to Ginza hotel
Day 11-14: Tokyo area (Ginza, Asakusa, Shibuya, Odaiba, Disney)
Day 15: Fly out from Narita

Thank you!

You can get the most out of 7-day pass by lumping your Kyoto/Osaka/Hakone trip in sequence.

For example:
JR Pass use day 1 thru 3: go to and stay in Kyoto
JR Pass use day 4 thru 6: Osaka/Hiroshima/Miyajima
JR Pass use day 7: travel to Hokone/Fuji (and recommend to stay there)

If you stay in Hakone, you will need to buy a regular train ticket back to Tokyo. But the distance from Hakone to Tokyo is shorter, thus price not too expensive.

For your travels within Tokyo, I recommend 72-hour subway pass (you can buy two of them). Use of JR Pass just within Tokyo is kind of a waste.

Both would pay off, but as said you get more value if you fit all the long distance travel into 7 days.
For Hakone, you can save money on a Hakone Free Pass.
http://www.japan-guide.com/e/e2358_008.html
http://www.japan-guide.com/e/e5210.html

Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are terrific.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jkgQdiQlK2bcKCzr2pJQ...
On another evening, you could also go and see Kobe.
Instead, take at least half a day and go see Nara.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jGyApXWSTIKwkioxuAKw...
For your Day 8, to add more value, you could also go to Himeji and see Japan's best castle.
You can fit in the best of Hiroshima & Miyajima, but get an early start. Aside from the Itsukushima Shrine with the giant "floating" torii, the Daishoin Temple is also a must-see place. There are several other pagodas and temples, and climbing Mt. Misen offers a fantastic view over the island and whole area.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-iCNZ6AdsKGUB-HPKNxP1...
Just beyond Miyajima is also Iwakuni, which is another picturesque place with its iconic bridge and mountaintop castle.
https://www.youtube.com/watch?v=0COOZXi66mM

For a cheap way to Narita, use the 1000 yen bus.
http://accessnarita.jp/en/home/

All of the long distance stuff is already in 7days (day 4 to 10), so only a 7day pass required.

I would be cautious about putting Hiroshima+Miyajima and Hakone+Fuji 5 Lakes on the same days.

While they can be done together. Both days would be a rush, and you'll wish you didn't do it that way.

I'd agree with the earlier response about visiting Osaka less. This is good because you can then add days for each of the above areas. And still fit all the long distance travel into 7 days.