Please also be aware that for Hakone, the ropeway and Owakudani area is closed due to volcanic activity. Substitute buses are running to bypass the area, but if it being closed would ruin your trip there, you also might consider going to the Fuji 5 Lakes area on a Fuji Hakone Pass.
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 wonderful.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha would be a tragedy. Nara Park has a lot of other great places, such as Kasuga Shrine, Kofukuji Temple, and feeding the many deer in the area. A few other great places missed out even by those who go to Nara though are the Isuien
https://www.youtube.com/watch?v=kk-7IoLELio and Yoshikien Gardens. They are gorgeous and not crowded at all.
In Kyoto you could spend weeks there and not see everything. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails.
Himeji Castle is also worth seeing.
In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain.
All in all, a lot more to explore and enjoy.