If you have 5.5 days, your grandmother with you and you all want to travel together, I would not recommend Kyoto let alone Nagasaki. Admittedly Kyoto is 2h20 from Tokyo using Nozomi (the fastest bullet train), you have to leave your hotel, go to the bullet train station, go through the crowd to the right platform and take the train which may take at least half a day to reach Kyoto. It is obviously up to you to make that decision - I would not hesitate a second if it was only you as just travelling on the train can be a cultural experience.

As other posters suggest, I would recommend side trips or walking around in Tokyo. Rather than worrying about how to get there, etc. why don't you rely on tours with English speaking guides? Do check the JTB (Japan's biggest travel agent) owned Sunrise Tours to look at the options. They offer lots of 1-day trips within Tokyo or to Nikko, Hakone or Kamakura. I am not affiliated with them but sometimes, when you're there for the first time, it's one of the most convenient ways to go around isn't it?

Please also check the JNTO 's website to get travel tips and also get ideas on what would interest you.