Do I need JR Pass?

Hi there!

I made a small list of ticket prices below to see if a JR Pass would be

Tokyo - Hakone ¥ 3,640
Hakone - Takayama ¥ 12,410
Kanazawa - Kyoto ¥ 6,710
Kyoto - Mount Koya ¥ 540 of the trip is covered.
Osaka - Hiroshima ¥ 9,950
Hiroshima - Miyajima ¥ 570
Miyajima - Nagasaki ¥ 13,640

For a total ¥ 47,460, the 14 day JR Pass is ¥ 45,100 so you would already be making savings. Note that I left the last part out since I don't know yet what you will do exactly. Personally I think Himeji may not be worth visiting at this point. The castle restoration is in the last phase however it will be closed for general public for this period. No matter where you go however the JR Pass will allow you to make good savings, for instance a one way Nagasaki - Osaka alone is ¥ 18,480.

Hope this helps!
Daniel