Please help with Japan itinerary

Add at least one more night in Kanazawa. One night does not even give you a full day.

I would route your trip using websites like these:
https://world.jorudan.co.jp/mln/en/?sub_lang=nosub
OR
https://japantravel.navitime.com/en/area/jp/route/

You can see the travel time and transfer requirements for the cities as you currently line them up. (or will you drive?) The travel time may cut into the touring time you have allowed.

Without knowing your interests or touring must-sees, I wonder about some of your allocations. 1 night in Kanazawa isn't a lot in my experience. (edit - cross-post with above. We agree!!) Kyoto could be a month's long stay! Consider close day trips like Nara, Himeji for the castle, Osaka if you like street food, Fushimi Inari, Arashiyama, there are others that work well from a Kyoto base. Quite a bit of time in Tokyo - consider day trips to Kamakura, Nikko or possibly reallocating some nights - long stay at trip beginning and 1 night prior to airport departure at end????

I don't see 2 nights in Kawaguchiko on most people's first time itineraries. It can be done as a day trip from Tokyo - many bus tour companies or DIY.

You are missing Hiroshima and MIyamjima Island which are included for many on their first trip. Hakone is much more easily routed than Kawaguchiko as Hakone is between Tokyo and Kyoto making an easy stop.

Not my trip, but I would consider this general loop, especially as a first trip hoping for more direct trains:
Tokyo, Hakone, Kyoto, Kanazawa, Tokyo
as a general direction with days added for trips from Kyoto and another night for Kanazawa or throw in Hiroshima.

Kanazawa is a nice destination but a bit out of the way compared to the other cities on your list. I am planning to be there in October and allowing a bit more time as well - several days to visit the city and the Noto Peninsula. The Kenrokuen Garden is a wonderful attraction and so I am allowing time to visit it during the day and then a second trip in the evening to see the illuminations which take place on weekends only, late October and early November.

Osaka is a short trip from Kyoto and may be worth considering. Lots to do and see there.

My experience is fairly Kansai centric, but here are some thoughts.

It looks to me like you have a lot of time in Tokyo. I don't know the city well enough to know if you need that amount of time.
There is a small ancient city called Nikko. It is an easy day trip from Tokyo. We enjoyed it. Take a look, it might be interesting to you.

Agree that two nights in Kanazawa makes a lot of sense.

Kyoto is wonderful. It is a bustling city in between all the tourist sights. Take a look at the Philosopher's Path. From memory, it takes you to some very nice, out of the way sights. Two of our favorite day trips are easy from Kyoto.

Nara Park is in the city of Nara. I expect you can get a walking map online or at the train station. Walking the loop around the whole park will help you feel like you are in old Japan. Feed the deer!

Himeji. The Himeji Castle is stunning and much better than some of the others. In Himeji, there is also a very rustic temple on Showa Mountain. Parts of the Last Samurai were filmed there.

My wife LOVES pottery. If this is a strong interest for you, look up Japan's ancient pottery towns and see if any of them make sense.

You might want to take a flight to another part of Japan with your open days. Perhaps either Nagasaki or Sapporo.

This sounds like a great trip.

Enjoy

Our itinerary was Tokyo - Shirakawa go - Takayama - Kanazawa - Kyoto - Mt. Koya - Tokyo. All good, but highlights were Tokyo, Kanazawa, Kyoto and Mt. Koya. Highly recommend a temple stay and nighttime walk through Okunoin Cemetery in Mt. Koya. Amazing!!

I fully agree with adding more time to Kanazawa. Once you make all the effort to get there, you should spend more time enjoying it. If you add to Kanazawa and Kyoto, you don't really need another three night stay. But maybe consider Nagoya as a possible destination. Or, drop Kanazawa (although I do love it) and add something else on your route, like Nagoya.

Given that you are devoting 8 nights to Tokyo and only 4 to Kyoto, perhaps you are more interested in modern Japan. If that is true, consider adding time in Osaka. If that isn't true, take time from Tokyo and add it to Kyoto to allow for day trips to Nara and Uji (could be done as a single fast day trip, but you'll miss a lot in Nara that way).

With 8 nights in Tokyo, you have time for day trips. Hakone could be done as a long day trip. Other rich choices include Kamakura, Kawagoe, Yokohama, and Nikko.

Final thoughts?

Tokyo (5 nights)
Tokyo explorations (Asakusa, teamLab, denim, Shimokitazawa, etc).

Kanazawa (2 nights)
Kenrokuen, Samurai District, Geisha District.

Takayama (2 nights)
Shirakawa-go stopover + Takayama Old Town; optional Okuhida onsen/ropeway.

Kyoto (4 nights)
Central/North, Arashiyama, Ine no Funaya, Uji+Nara, Southern Kyoto.

Kinosaki (2 nights)
Travel Kyoto → Kinosaki (2.5h). Yukata stroll + onsen-hopping. Next day relax, explore town fully (maybe Amanohashidate side trip).

Tokyo (3 nights)
Kinosaki → Tokyo (≈5h). Evening in Tokyo.

• Tokyo- Fuji day trip (Kawaguchiko: Lake, Maple Corridor, Chureito).

• Tokyo- Kamakura + Enoshima (or Mt. Takao foliage if we want nature/fall foliage).

• Tokyo → Flight Departure day.

Consider a day trip from Tokyo to Nikko, north of the city, its amazing.
Osaka is worth a day.

Any recommendation on number of days spent?

More days in Kyoto?

Caveat to below - I am a "slow" traveler preferring to see several places in more depth rather than many places for a couple hours. That may not be your preference.

Not sure about your Kyoto plans, but I don't know your particular interests (no need to provide unless you want specific suggestions.)

Kyoto - 3 full days Many would suggest more as there are so many options. Specifically for your plans:

Ine no Funaya is not in Kyoto - no where near. 2+ hour train ride to Amanohashidate Station, then a bus ride. Over 3 hours one way. That would be one full day - even if you find a day tour rather than DIY, it will be a day long venture.

Nara + Uni - doable with a long, full day

Arashiyama and exploring Kyoto - 1 full day?? That shorts both in my experience. Iots of options in Arashiyama as well as various areas of Kyoto that aren't temples.

If you enjoy vibrant neighborhoods (I'm looking at your Tokyo plans), Osaka's Dotonbori and Shinsaibashi-suji Shopping Street (adjacent) might be of interest. Osaka is commuting distance from Kyoto, short train ride, many options for routes. This area of Osaka could be visited late afternoon or early evening. Osaka is a vibrant city, very different "vibe" from Tokyo to my way of thinking.

Kanazawa to Takayama - I assume you have looked at the bus schedule. https://www.nouhibus.co.jp/highwaybus/kanazawa_en/
Agree that the stop over in Shirawago is well worth the time, I would buy the bus tickets in advance before leaving the US. They can sell out. (or may not, but I'm not a risk taker..... I've always had tickets before hand.)

Kanazawa - you might enjoy walking through the Omicho Market. I don't much like markets but found this one quite enjoyable - clean, covered, restaurant options were appealing. By contrast in Kyoto, Nishiki Market is generally so crowded that the experience is one of endurance just trying to walk through. Given your short Kyoto time, I would resist being tempted to visit.

Tokyo day trips - Nikko is truly special. Can you fit that in as a day trip during the days following your arrival? Beautiful natural area and the temple complexes are stunning.

Takayama was my favorite place in Japan. Enjoy it all!