Should I get a 7 or 14 day JR pass?

Hi,
We are travelling in Japan for 2 weeks as outlined below. Just trying to work out whether it is worth getting the 14 day pass or could we manage with the 7 day pass and a few extra trips?
Tokyo 12-16th April (5 nights)
Hakone 17-18th April (2 nights)
Kyoto 19-22nd April (4 nights)
Hiroshima/Miyajima 23rd April (1 night)
Osaka 24-25th April (2 nights)
We fly into Tokyo and out of Osaka.
Thanks!

7 day pass, 19-25 April

The answer may be neither. The JR seven day pass is 29110 yen.

You will need to look at the regional transportation pages to figure out if you need a pass from your airport when you arrive in Tokyo. Here are the links for Narita and Haneda:

http://www.japan-guide.com/e/e2027.html
http://www.japan-guide.com/e/e2430.html

Once in Tokyo, you may want to use an IC card for days with little trips, and day passes for days with several stops. The IC card will also work in Kyoto, Hiroshima, and Osaka.

http://www.japan-guide.com/e/e2359_003.html

A JR Pass is not very effective inside of Hakone. I believe that Hakone is best done with a Hakone Free Pass:

http://www.japan-guide.com/e/e2358_008.html

If you have an active JR Pass, you could save 1140 yen, and purchase the area pass at Odawara station. If you don't, the cost of a one way Limited Express ticket from Shinjuku Station to Odawara Station is 1770 yen, so I would probably get the full package from Shinjuku Station.

You will then need to get from Osawara Station (Hakone) to Kyoto. I am going to suggest buying a discount ticket:

http://www.jrtours.co.jp/kodama/en/?scid=jtt_kepEng001

From the breakdown on the link, you may have to buy a ticket for Shin-Yokohama to Kyoto, which will cost 9900 yen.

You will then use local busses for the most part in Kyoto. Purchase a JR Kansai Hiroshima Area Pass and activate it for the final five days of your trip (April 22nd-April 26th). This will cost 13000 yen.

http://www.japan-guide.com/e/e2361_kansai_hiroshima.html

On your last day in Kyoto (April 22) use the pass to go south to Nara, Byodoin Temple and the Inari Shrine, or possibly the western Kyoto area (Bamboo forest, Arashiyama). Both of these can be accessed with a JR train Pass.

Go to Hiroshima, Miyajima, return to Osaka, and get to the Kansai Airport with the JR Pass.

The cost of the single discount ticket and the JR Hiroshima pass is 22900 yen, compared to the 29110 yen for the JR Pass. There are pros and cons to going either way:

The discount ticket is on the slower Kodama Shinkansen train, which you might not want to ride on.

The benefit that you will see from the JR Pass is making seat reservations. The Hiroshima pass is non-reserve only, but also includes Nosomi Mizuho Shinkansen trains.

The Hiroshima pass also does not cover Shinkansen between Kyoto and Shin-Osaka Station (very weird, in my opinion).

The JR seven day pass will run out on the 25th, so you will also need to pay for transportation from where you stay in Osaka to Kansai Airport, which is covered with the JR Hiroshima Pass.

This is your trip, and I am just trying to show you some options. The choice is yours.

Wow! Thank you for such a comprehensive reply!

Definitely not a 14 day Pass.
Use www.hyperdia.com and think in Yen: uncheck the Nozomi etc box, and the Private railway box
a 7 day Pass with trips around Osaka Loop line should pay off, even in terms of convenience:
you can reserve seats for free on the Shinkansen;
you can use it on the JR Ferry to Miyajima;
if you use it from 19th, from Odawara to Kyoto, then on to Hiroshima etc , and also out to KIX on the 25th, then it should be a comfortable and viable option.