When to start JR pass

We fly into Narita then spend 4 days in Tokyo including a day in Kamakura. We then go to Nagoya and Inuyama for a night, Takayama for 2 nights then back to Nagoya for a night before going to Kyoto for 3 nights with a day in Himeji and 2 nights in Osaka with a day in Nara. At the end we return to Narita (I know Kansai would have been better but didn't work as have a free return ticket) Since we are there for 15 days should we buy a 14 day JR pass and pay for the initial trip to Tokyo or pay the extra and get a 21 day pass? Any other suggestions for travel?

Note that JR Pass makes sense only for long-distance travel. Meaning, you don't have to have JR Pass for the first four days in Tokyo, including a day trip to Kamakura. It would come out cheaper to buy individual tickets, or (for convenience) use prepaid card.

I can't quite figure out the itinerary you've got (can't quite tell when you say "with a day in..." you mean an extra day or not), but you might want to calculate the fare for individual trips, and check if the price justifies any pass - even single tickets might come out cheaper.