I agree with rkold on removing Kawaguchiko entirely from your travel itinerary.
Are you wanting views of Mt Fuji? If so, then you can get very good views (on a clear day) of Mt Fuji from Owakudani [Owakudani:
https://www.japan-guide.com/e/e5203.html] .
If you do the classic Hakone Round Course [
https://www.japan-guide.com/e/e5210.html], then Owakudani is a part of the route.
I would also personally remove Kawaguchiko, so that you can have an extra day in either Kyoto or Tokyo. As what you want to see, would be very fast paced and not an enjoyable experience.
Day 1: Sensoji Temple, Nakamise Shopping Centre, Tokyo Skytree & Ginza.
Day 2: Imperial Palace, Shinjuku Shopping [I'd also recommend checking out Shinjuku Gyoen] & Shibuya.
Day 3: Tokyo sightseeing [Might even recommend visiting Ginza on this day instead of Day 1]
Day 4: Hakone [I recommend doing the Hakone Round Course]
Day 5: Travel to Kyoto, Sanjusangendo, Kiyomizu-dera Temple & Fushimi Inari Taisha
Day 6: Kinkaku-ji Temple, Tenryu-ji Temple, Sagano Bamboo Forest
Day 7: Day trip to Nara [from Kyoto]
Day 8: Day trip to Osaka [from Kyoto]
Day 9: Day trip to Osaka [from Kyoto] or Kyoto sightseeing.
Day 10: Leave Osaka.
Depending on where you are staying in Tokyo and Kyoto, these days are more evenly spaced out and more easily doable than what you originally had proposed.
As for Hakone, are you wanting to stay at a particular ryokan or hotel?
If not, then make it a day trip from Tokyo. The Hakone Free Pass will give you a round trip journey between Shinjuku Station and Odawara station (on the Odakyu railway). This pass will also give you unlimited access to the various transportation within the Hakone area. And this is great for those travellers that explore Hakone by following the classic 'Hakone Round Course'.
[Hakone Free Pass info:
https://www.japan-guide.com/e/e2358_008.html]