After church, Merton finds that dusk has fallen. He goes to Marian's place to see if Kate is there. He finds the house "ugly almost to the point of the sinister" (188.8.131.52). That's not exactly a ringing endorsement of the people living within.
When he sees Kate, he says he's there to tell her that Milly has died. Kate isn't so concerned about this as she is about whether Merton has heard word from Susan Stringham. Merton doesn't know why this is an issue, but lets it go.
Kate mentions that some trustee guy is coming over from America to look after Milly's estate and fortune. Merton is happy about this, since it'll take pressure off of Susan Stringham to make the funeral arrangements.
Back to the question at hand, though: Merton says he went to Sir Luke's on his way to Lancaster Gate, because there was something he needed to tell Kate. He says that he has made a decision about something.
Merton says that he has received a letter from Milly that was postmarked very close to the day of her death. But he realizes that she wrote the letter sometime before then and then delayed sending it until the very end of her life. It's definitely in her handwriting, based on the address she's written on the envelope.
He pulls out the unopened letter from his pocket. Before handing it to Kate, though, he asks her one last time how Lord Mark knew that she and he were engaged.