Hi,
Well it looks like it is shaping up nicely.
As I'm sure you're aware, July is summer and it will be very sticky and muggy - bring enough sunblock and deodorant, and be sure to keep hydrated. You are putting some generous time into Kagoshima and Nagasaki. If you want to do more for Kagoshima, you could go south a bit to Ibusuki and enjoy the volcanic sand baths. It's also a bit long to go from Kumamoto to Nagasaki, and if you want to see a real gem along the way, visit the Yutoku Inari Shrine along the way - you'd be taking the bullet train to Shin-Tosu and then the Kamome Ltd Express either way, the only change is getting off at Hizen-yamaguchi Stn to get a local train to Hizen-hama Stn, and then take a short bus ride or just grab a taxi.
For Nagasaki, you can certainly see all the good sights, and if you like, take the 3 hour tour to Gunkanjima or Battleship Island. It is something you will never forget - like going to Chernobyl without the radiation. If you want more to see, I mentioned Unzen, but there is also Huis Ten Bosch and [the Sasebo Island Cruise]. Within Nagasaki City get a streetcar day pass to see most of the sights. There is a tourist info office in Nagasaki Stn and worth a visit for maps and info.
I'm not sure what the drive is to see Osaka; every city has something of course, but Kyoto and Nara outshine the place. If you have not seen Himeji with Japan's best castle, that is also worth your attention. At night Osaka comes alive and has some great places to see though, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are wonderful.
Keep some aspirin and eye drops on you for the long flight - you'll need them by the time you land.
Good luck.