I believe that this question is asking for information that can all be found in chapter 7. Near the end of chapter 6, Kit and everybody else is shocked to find out that William Ashby wants to come and call on Kit. It is shocking because he knows next to nothing about her; however, Kit is the new exotic girl in town, and his attraction makes sense.
Chapter 7 begins with Kit and William sitting alone together in a room by the fire. The rest of the family has moved themselves into another room, and the entire encounter is incredibly awkward. William and Kit find next to nothing to talk about, and Kit is relieved when John shows up and the rest of the family enters to engage the new guest in conversation.
As the chapter begins to wrap up, readers are told that William's calls fell into a regular pattern. He would come and visit every Saturday evening. The conversations did not necessarily get any better, but Kit is smart enough to think about her future welfare. She believes that a future life with William is possible. She knows that it would ultimately be a boring life in which nothing is expected of her, but that sounds somewhat appealing in light of the fact that she is currently working herself to the bone.
Kit also admits that William is indeed a good-looking guy. Add that to the fact that he is wealthy. In terms of future security, Kit knows that William is a good choice for a future husband. Kit just doesn't get the stereotypical "warm fuzzies of love" with William, but the chapter ends with the narrator telling readers that Kit began to look forward to her Saturdays with William.