Flash back to just before Milly saw Merton and Kate at the National Gallery. Upon getting back from New York, Merton meets with Kate and asks her to hang out. The plan is for him to come over to Lancaster Gate that Thursday. But instead of waiting until then to see each other, they decide to go to the National Gallery, where no one will know them...
It turns out that Kate had known all along that Merton and Milly have met before. Merton said so in his letters to her. Thus, Kate is no doubt curious about why Milly has been so quiet on the subject of Merton.
After lunch at Milly's, Merton puts Kate into a cab and goes for a long walk by himself. We finds out that during their time together that afternoon, he asked Kate whether Aunt Maud knew he was back. Kate says Maud wouldn't have heard so from her, but that she'll tell her right away that he's back.
Kate then tells Merton to come over to see Aunt Maud, who will no doubt allow him to be alone with Kate at some point. She also mentions that they're leaving town soon and that she has a plan that might help them get married.
As Merton goes for his solitary walk, he tries to mull over what Kate has told him and can't make sense of it. Basically, he figures that Milly will be an ally to him and Kate and will help them get some time alone together.