Just to add to the list from Remoter33, my brother and his wife stayed at one there several years ago, the Asaba ryokan. It was fantastic but stupidly expensive.
I just checked and prices are upwards of £1,000 per night half board for two people in one room.
https://www.tripadvisor.co.uk/Hotel_Review-g790340-d304281-Reviews-Asaba-Izu_Shizuoka_Prefecture_Tokai_Chubu.html
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"...and a private in-room onsen!"
They are going to be more expensive than ordinary basic rooms. Some ryokans don't have any rooms at all with in-room onsen.
"Yukairou is famous but is expensive."
I don't think that it is so expensive (but that's just my point of view, it all depends on the budget you have available). Half board for two people with a private onsen in the room next September starts from £325 depending on dates.
But you can also get much larger rooms, such as their "Detached Annex Cottage / Japanese-Style Room (53 m²)" from £385.
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If I may make a couple of suggestions a little further afield?
About 5 miles to the south of Shuzenji there is a small town called Tsukigase. There is a bus to there from Shuzenji but it's a lot easier to just take a taxi.
There is an excellent ryokan there by the name of Sagasawakan. A room with a private onsen which is half indoors/ half outdoors is around £345 per night half board.
https://selected-ryokan.com/ryokan/sagasawakan_izu-shuzenji_shizuoka_tokyo-suburbs.html
There is also another lovely little ryokan there called Tsukigase Onsen Ufufu where each room has its own private onsen outdoors and most are in a small garden, again from around £350 per night half board.
https://www.ikyu.com/en-us/00001831/
Photos of the private onsen in the garden attached to the room:
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