Juliet writes Sophie that Sidney is getting too important, given that he's sent Billee Bee to collect the letters. Billee Bee was very seasick upon arrival but has been a good sport. Kit, unfortunately, doesn't like her and refused a kiss. Juliet asks Sophie how she corrects Dominic when he's rude like that.
Just as with the romance between Dawsey and Juliet (which many don't see), it's important to recognize that Juliet casts Kit's dislike of Billee Bee as rudeness and not of a signal that there's something off.
Juliet confides that she's been worried about Kit since she learned of Elizabeth's death. She's decided to speak to Mr. Dilwyn, Kit's legal guardian, about possibly fostering or adopting Kit. She's not sure if Mr. Dilwyn will agree to adoption, given that Juliet is unmarried and without steady income. Juliet hasn't told anyone yet; she's too afraid of what Amelia and Dawsey would think. She wonders if Kit would even agree to go to London.
Now that Juliet has had several months to grow close to Kit, she recognizes that the most useful and profound thing she can do for Kit is to provide her stability and legitimacy by trying to adopt her. This would mean that Kit's future isn't in danger, as she'd have a dedicated legal guardian to rely on.