With eight nights, and a plan to return in the future, I would find an Airbnb or similar weekly rental and spend all eight nights in Tokyo, with day trips to Kamakura, Fuji and maybe Nikko (though Nikko is better as an overnight).

Kamakura and Nikko will give you a taste of Kyoto (and they're frankly less crowded anyways, at least during the week).

I actually leave for Tokyo tomorrow to do just this with a group of friends. My seventh visit to Japan, their first but we only have 7 nights. We're spending the entire time in Tokyo with day trips to Kamakura and probably somewhere up in the mountains a bit (TBD).