For two?

I cannot think of one Ryokan within Tokyo or Kyoto where you would get a great Ryokan experience for ¥20,000, never mind ¥10,000 (Ryokan prices are generally per person)

The only solid suggestion I have for you is to go to Kurama Onsen on the outskirts of Kyoto. It's just 30 minutes and a ¥410 train trip from Demachiyanagi station in Kyoto - so no need to compromise your itinerary.

Or reduce your hotel budget for other nights within Tokyo and Kyoto and splurge out on one of Kyoto's better establishments - like the the Hiiragiya or Tawaraya.

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You could go to Andon Ryokan or Kimi Ryokan in Tokyo if you just want the experience of sleeping on a futon or stay at the Motonago (no curfew) or Three Sisters Inn in Kyoto (which has a curfew)- the last two would probably offer perfectly decent experiences and a service level comparable to a good Minshuku (Japanese style Bed and Breakfasts - although dinner is often served at Minshuku) rather than a good Ryokan.

But what I would certainly do in your place is allow one late afternoon/early evening on your itinerary and go stay at one of Hokkaido's extraordinary Onsen Ryokan instead.