Hi,
Your itinerary, as is, simply would not work. From Tokyo to Nagasaki by train using the pass would be over 7 hours, almost completely eating up your daytime. What you could do however is replace Nagasaki with Kanazawa, which is a beautiful city in itself, with the country's most beautiful traditional garden and many other nice sights.
If you traveled by bullet train as much as possible in the early morning and after sunset, you'd still be limited in what you could see but still manage to see a few places. In Tokyo 2 of the most popular places are the Sensoji Temple and the Shibuya Stn Hachiko crossing, where an ocean of people are always crossing the street.
In Kyoto the three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming.
For a day trip to Hiroshima/Miyajima, it is possible if you start very early - you can cover the best places in Hiroshima City (Peace Park/Museum, castle, Shukkeien Garden) and spend about ¾ of a day to see Miyajima. Some people just go see the Itsukushima Shrine with the giant "floating" torii, but that'd be really cheating yourself - the Daishoin Temple is also a must-see place. There are several other pagodas and temples, and climbing Mt. Misen offers a superb view over the whole area when the weather is good.
And if you can squeeze it in, seeing Nara, especially the Todaiji Great Buddha is an extraordinary place, as well as many other historic temples, shrines and gardens in the area.
Best of luck.