In Kinosaki Onsen, we stayed at "Nishimuraya Honkan". It's a typical traditional Japanese hotel (ryokan) and has a lovely Japanese garden. The garden and the room was somehow smaller than I imagined but still very nice. Both my husband and I prefer these traditional and old style ryokan when we go to hot spring and my husband wants to sleep on futon (it's different from what western people call "futon" in western countries...). These days, even these typical Japanese ryokan tend to have beds rather than futon (because Japanese people also prefer beds in recent years specially older people) so our choices tend to be limited.
Anyway, we enjoyed staying in this place. We saw some western people there, too, and their website is in both English and Japanese so I guess it's known to non-Japanese.