Advice on Ryokans and good sites to search/find/book an Onsen Ryokan? Hi all,

So my gf and I are flying into Narita in mid June for 12 days. We have plans to spend most of the time in either Tokyo or Kyoto, but have 2 nights where we want to experience a Ryokan. We're getting JR railpasses, so we are more than willing to travel, especially to an onsen town, as long as its relatively close to a train station, but are also ok to stay at a Ryokan that's in/very close to Tokyo or Kyoto. Our budget is $250-400 per night, have some legit Kaiseki, and we would like to have private or coed baths, but thats not a top priority. I've been looking at areas like Hakone, but I'm kind of getting overwhelmed. Is there any booking site that only searches for Ryokans? Are there sites that search for Ryokans in all of Japan? Sorry if my question is redundant or stupid but it's our first trip to Japan and it's also been kind of last minute too. Thanks everyone!