21 days rail pass, is it worth it for us?

Hi there and good morning from Europe,

First I think this makes great start to explore Japan.

Tokyo Nikko ¥ 10,860
Tokyo -> Komagane ¥ 7,130
Komagane --> Takayama ¥ 11,280
Takayama --> Kanazawa ¥ 5,840
Kanazawa --> Kyoto ¥ 6,710
Kyoto -->

So far you are already at ¥ 41,820 a 21-day JR Pass is ¥ 15880 away. Which will be easy to reach, for instance a return to Tokyo alone from Kyoto is ¥ 13,220, which in turn just leaves ¥ 2,660. Calculations aside I do think that a 21 Day JR Pass will quickly become a good option.

Where to go from Kyoto?

With 5-7 days left, there are a couple of options here, you can go down to visit Hiroshima and Miyajima. Nagasaki is also not too far away from Hiroshima. Instead you could also visit Matsue and Izumo Taisha from there, before making your way back to Tokyo.

You could also stay a little longer around Kyoto and make a couple of day trips from there, great destinations include Nara, Kobe and Himeji.

Let me know if you have any special interests and I can make some more specific recommendations!

Hope this helps,
Daniel

Side note: If you really wish to visit Yakushima, you can take the train down to Fukuoka and fly from there with JAL they offer a package called Yokoso Japan for foreign tourists and flights can be as low as 10,500 for one way. This can be a little tricky as it has to be booked in advance and some limitations apply.