The fact that Kyoto does not have many onsen, especially natural ones, makes your task difficult.Kurama has an onsen ryokan for a long time, but I don't think they have private baths:
http://www.japan-guide.com/e/e3933.html
In recent years, boring machines have provided the Ohara and Arashiyama areas of Kyoto with some hot spring water and ryokan, but I am not familiar with individual establishments.
About Ohara:
http://www.japan-guide.com/e/e3932.html
About Arashiyama:
http://www.japan-guide.com/e/e3912.html
Near Kobe is Arima Onsen, one of Japan's most famous and oldest hot spring resorts. There you should find quite a few ryokan with private baths. But it is more than an hour outside of Kyoto:
http://www.japan-guide.com/e/e3558.html