Personally I'd bin off Osaka if you only have a few days in Japan. Kyoto is well worth a visit as is Nara if you have the time. That said, Osaka is right next door to Kyoto, so...
It's not my part of Japan so I can't give too much info but Osaka, Kyoto and Nara are all near to Himeji. Hiroshima is pretty close too and if your down that way should definitely be visited. Out of the four, Kyoto and Hiroshima are musts, Nara and Osaka bonuses.
Oops, forgot to quote...
If your daughter is fine with the long haul flight and jet lag(I find going to Japan to be much worse than coming from it) then I say do it.
I can heartily recommend Asakusa as that's where I live. However, for a tourist it might be easier if you headed a kilometre or so west and stayed in...