Juliet tells Sophie that she finally went to ask Amelia her thoughts on adopting Kit. Amelia was relieved and both women cried. Then, Amelia said she'd go with Juliet to see Mr. Dilwyn, as she's known him since he was a boy and he wouldn't dare refuse her.
Because Amelia has a long history with Mr. Dilwyn, he's also a part of the family and therefore, will trust Amelia. Amelia's happiness that Juliet wants to adopt Kit shows that Juliet is now a real family member.
Then, the best thing happened: Kit woke Juliet up in the morning and silently offered Juliet her box of treasures. The box contained a baby pillow, a photo of Elizabeth, a handkerchief, a signet ring, and a book of Rilke's poetry that Christian gave to Elizabeth. In the book was the note that Elizabeth left for Amelia on the night she was arrested. Elizabeth's medal was there too. After she'd looked at the treasures, Juliet held open her arms. Kit climbed in, crawled under the covers, and went to sleep. Juliet knows now that she wants to stay in Guernsey to raise Kit. She says that Amelia believes Mr. Dilwyn will likely grant Juliet guardianship, even if he won't grant adoption.
In particular, the presence of the medal indicates that at some point, Eli passed it onto Kit—and likely told her the same story about it being magic that Elizabeth did. In this way, he paid forward Elizabeth's kindness to him and allowed Kit to connect with her birth mother. Crawling into bed with Juliet and sharing her treasures shows that Kit is now fully committed to Juliet and trusts her with her secrets, like she would a parent.