I have been below two and liked a lot.
- Konya (紺屋) in Kagurazaka
I'm not sure if this is a type of kaiseki you are looking for. They offer multi-dish course and also a la carte menu. Maybe rather casual type of kaiseki. They have a good selection of sake... you can try sake from all over Japan.
The restaurant renovated an old house and has a small Japanese garden... atmosphere is brilliant.
- Washoku Shimizu (和食 清水) in Shinagawa
The restaurant above is inside Shinagawa Prince Hotel Shintakanawa. I had a dinner at a private room of this restaurant. The dish was beautifully decorated by artisan cooks at the hotel. The content of the course varies depending on the day, but the staff prepared an English written list of dishes in the course when I asked.