Not that it cant be done but its going to be more time on the train than actually sightseeing.

My opinion is that if you really want to travel to these places, then uproot from tokyo and stay there.

If you so decide not to travel there then the 14 day JR pass dun seem to be a worthwhile investment. I think there are other JR passess that suit travel in and around Tokyo better and at a much less price.

I do not recommend a day trip to Kyoto (though it can be done), definitely not to Hiroshima.
There are one-day bus tours to Mt. Fuji area, so it may be OK. Nearby Hakone can be done in a day trip from Tokyo too - you can use JR Pass up to Odawara (served by JR trains), then take non-JR means of transportation within Hakone, as there is no "JR" trains within Hakone area.

For Kyoto and Hiroshima, I recommend at least one overnight stay at each place; for Kyoto, you might want more days.
Even for Mt. Fuji/Hakone area, a night's stay there at a hot spring inn would be nicer :)

Your hotel's name does not include "Narita" or "Ginza," right?
Then your hotel must be Hotel Nikko Tokyo, which is located in Odaiba area.
This Hotel is connected to Daiba Station of Yurikamome. No JR line is available in this area.
You can use a prepaid IC card (Suica or PASMO) for fares in Tokyo of Yurikamome, JR-East, Tokyo Metro, Toei subway and some other railroad operators.

Daiba --[Yurikamome: 310 yen, 15 min]-- Shimbashi (West Exit) -- ;

- For Kyoto: [transfer to JR]-- Shimbashi (Track 1) --[Tokaido Line: within 5 min]-- Shinagawa --[Tokaido Shinkasnen: 2hr 40 min by HIKARI]-- Kyoto.
- - For Hiroshima: Kyoto --[Rapid / Special Rapid: within 35 min]-- Shin-Osaka --[San'yo Shinkansen: 1hr 30 min by HIKARI]-- Hiroshima.

- For Ueno (Ueno Park): [transfer to JR]-- Shimbashi (Track 5) --[Yamanote Line's inner loop: 10 min]-- Ueno (Park Exit).

- For Asakusa (Kaminarimon, Sensoji Temple, etc.): [transfer to Tokyo Metro]-- Shimbashi (Track 2) --[Ginza Line: 20 min]-- Asakusa (Exit 1 in daytime).