1 Free Day - What to do?

Hey all,

Wife and I are traveling to Japan in late October - early November and I'm trying to plan around the big cultural sites and see the colors of autumn in.

I've built out my travel itinerary for my wife and I like so:

Arrival + 1 day in Tokyo

2 days in Nikko

1 day in Hakone with a stay in a ryokan onsen

4 Days in Kyoto/Osaka or Nara

2 Days in Takayama

1 Free Day

2 Days in Tokyo with flight out at the end

1. I'm trying to figure out what that free day can turn into and I'm not sure if: 1. Its worth delaying Takayama a day to do an extra one for Nara or more time in Kyoto or Osaka
2. I should spend the extra day in Takayama
3. Overnight stay in a place around Lake Kawaguchi - thinking Hakone may already cover that
4. Onsen night near Ito before heading back to Tokyo

I'm totally open as long to suggestions here too. If there's something worth going to, please do let me know!