Since Kaneki's hand stopped at the number 20 on the calender, he prob was thinking about his and Arima's birthday. Honestly, Eto's declaration that she's not the One-Eyed Ghoul King while showing Kaneki dozing over the calendar can only refer to Arima in my eyes. Can't be Kaneki cause he's not 100 yrs old, lol.
Who said that this King existed anyway? My guess is that Eto trying to create one, and now she chouse Kaneki.