According to the third URL below, Kounoyu (こうの湯), a ryokan in the Kurokawa Onsen area, is the most popular onsen ryokan in Kyushu in the 1 to 10 room category. Looking at the website is making me want to save up some money (such facilities tend not to be cheap) for a visit. – Kirk
P.S. I’m linking to the English side of their site, which I think it pretty good. But, one thing I don’t like about it as that there’s no link from the English back to the Japanese (clicking on “English” is basically a one-way ticket). This is problematic in part because the Japanese side has a much better interface for checking for openings and making reservations. I see this as another example of the assumption that people who might want to look at an English-language website couldn’t possibly have any reason to (or be able to understand) the Japanese website. :(