JR Rail Pass - 14 or 21 day pass?

Can anyone advise if it's best to purchase the 14 day or 21 day JR rail pass given the below itinerary?

Day 1 - Tokyo (3 days)
Day 4 - Kyoto (4 days)
Day 8 - Osaka (3 days - side trips to Nara, Kobe, Himeji, Hiroshima, Miyajima)
Day 11 - Takayama (2 days - side trip to Shirakawa go)
Day 13 - Oku-hida (2 days)
Day 15 - Tokyo (6 days)

I wasn't sure whether the JR pass would be useful while in Tokyo itself? From my research, I believe it can't be use in the metro (that this requires a different pass to travel within Tokyo?)

I am flying in and out of Narita.

Thanks for your help!

I recommend to use a 14-day pass and use regular tickets (or a NEX&Suica package) for the first few days in Tokyo. It comes cheaper than a 21-day pass.

would we ever need our JR rail pass within Tokyo or areas around Tokyo?

Youwill find there aree many subways as wel as the JR city loop line, the Yamanote line. you should consider if you are doing many sidetrips from Tokyo, for example, are you going to Nikko or Kamakura, Yokohama perhaps? go to hyperdia to figure out the costs, and if you find it's heading toward the cost of a 21 day pass, you may prefer to get that for sheer convenience.