Did this exact trip last month with my two daughters ages 19 and 16. Flew LAX to KIX, spent about 4 days in Osaka, visiting Kyoto for a day, and Nara for a day. Nara was unexpectedly fun, because of the large number of deer roaming the area between the station and verious temples. Your kids will certainly get a kick out of that.

Took the bullet train to Tokyo and spent another 4 days there. Did the usual Tokyo stuff plus a day trip to Kamakura, highly recommended. Flew home from Narita.

Used several guidebooks, but found the United airlines three perfect days series most helpful.

Where to Stay
Had a couple of award nights at the Osaka Hilton. Great hotel. Small rooms, but wonderful exec. lounge.

Used a couple of award nights at the Century Hyatt in Shinjuku. It's across the street from the Tokyo Hilton. Shinjuku is a great area for kids. The hotel is a 10 minute or less walk to Shinjuku station and the hotels have a very frequent shuttle. You're on the "calm" (west) side of Shinjuku. Go around the station to the east side and you're in a completely different universe.

Shinjuku station has subways and trains to wherever you want to go, but unfortunately the bullet train from Osaka goes to the Tokyo station on the east side of the city, so you'll then need to train or cab it to Shinjuku.

Summer Heat
Yes, it was hot and humid. We did everything via train and walking, and the heat and humidity took their toll. Do not plan on doing more than one thing per day. We were wrung out at the end of each day.