Shinkansen travel with the JR Pass (14 or 21 days)

Hello,

My girlfriend and I are planning a 23-day trip around Japan in September, and we are wondering which JR pass suits our needs. Here is our itinerary:

Tokyo: 5th-12th
Hatsukaichi-shi (near Miyajima): 12th-16th
Kyoto: 16th-20th
Nikko: 20th-24th
Tokyo: 24th-28th

We have the feeling that the 21 day pass might be the most suitable, as it can be used on some of the Shinkansen travel and we are likely to take 2 day trips from Tokyo in the first week of the trip, to one of the Fuji lakes and perhaps the coast. On the other hand, it may be better value and easier to use a 14 day pass from 11th-25th, covering one day trip and use a Suica card to get around Tokyo.

Is it worth it to buy the pass for this week of travel around Tokyo?
Thank you!
Ben and Hazel