Actually if you are hoping to save some money and see Mount Fuji and Hakone. Try going down to Odakyu on your Day 2 visit to Shinjuku and check out the Fuji-Hakone Free Pass:
http://www.odakyu.jp/english/freepass/fujihakone.html
My girlfriend and I did this earlier on in June. Great way to travel Hakone imo. The bus, ropeway, cablecar, tozan line trains all included together with the 2 way romancecar tickets.
The price is only about 100 SGD and if you top up a bit more you can actually get to go on the RomanceCar. It brings you straight into Hakone and the transportation inside there will be all taken off by the freepass. Even the boat ride around the lake is included. From what I know, the Shinkansen does not actually stop exactly at Hakone-Yumoto. It stops 1-2 stations prior to that and trust me the JR station is quite far from the actual Hakone area.
And for Kyoto & Nara: there are some express trains in the morning you might want to pay a bit more to catch. Save you some time.
Here's a tip where you are at Kyoto go to the bus sightseeing counter in front of the station and buy a 1 day sightseeing tickets for 500 yen. Each bus rides cost 220 yen flat fare anyways and if you are travelling within Kyoto take a bus. The trains aren't that well connected as they are in Tokyo.
Places you might want to go in Kyoto:
Inari -> Remember Memoirs of a Geisha: The little sayuri running through all through Tori Gates to pray...
Gion -> The hanamachi
Ginkakuji
Kinkakuji
Nijio-jo
Heian Jingu Shrine
Nara:
See the shika(Deer)! Yes they are wild. Your daughter should like them.
One last thing: Japanese actually know their English very well. They are shy of making a fool out of themselves. So take the first step with some clumsy japanese and they will see that you need help desparately and will try and help you in English

All else fails try this: "Sumimasen, Onegaiishimasu..." aka "Sorry but could I implore you to help me..."
Hope you have a great time in Japan