Had a consultation with MrLapLap over this (by pushing me he has a better idea of what constitutes reasonable effort in regards to getting around in Tokyo).
Roppongi (i.e. the Tokyo ANA IC as well as the Grand Hyatt and others) is not a great area to navigate. Roads and pavements are difficult and the subway/metro is deep, deep underground and takes an age to access. It isn't impossible by any means but not an area we'd recommend to base yourself in.
Both the ANA IC Tokyo and, paricularly, the Sheraton Miyako were convenient for us as my father-in-law lives in Azabu Juban. Getting around Tokyo from either hotel wasn't nearly as straightforward as it was from:
Strings Intercontinental.
This hotel ticked all the boxes for us. However, options for a family here are pretty limited and there are no connecting rooms. Only way I could imagine it working would be to get a King bed corner room and request an extra bed for a child with the understanding that one of your kids would share the King bed with the adults.
The other thought to put in your head is an evacuation. A large earthquake may make the elevators inaccessible so getting down from or up to the hotel via the staircase is an eventuality that could happen. The Strings is pretty high up, views are good, this is definitely your call.
So... other options...
Shinagawa worked for us and I went back there last March with a 2 and a half year old. It took a while for her sleep patterns to adjust and she had a blast running riot in the adjacent Konan business district. No cars, lots of greenery, lots to explore, turned out to be a great place to have her exhaust herself in.
But you are planning on visiting in August so that's not a concern, the trick will be to NOT exhaust yourselves.
The Strings is VERY close to Shinagawa station. Other nearby stations aren't so close, so the two Takanawa Hotels (Grand Prince Hotel Takanawa and New Takanawa Prince hotel) that I often recommend here are not so ideal. What is a nice 20 minute stroll in the Spring or Winter could be quite gruelling in August, particularly as part of the route is uphill.
I still think the big question here is how you anticipate getting around.
What are your initial thoughts on this?
It will make a hotel recommendation much more meaningful.