Brainstorming with Hats

Jane, Kate, and Lily are perfect logicians and always tell the truth.

They are each wearing a hat marked with a distinct positive integer. They know that one of the hats has a single-digit integer, $N,$ written on it, and that the other two hats each have a divisor of $N$ written on them. Each person can only see the two numbers on the others’ hats.

• Jane: "I don't know my number."
• Kate: "I don't know my number."
• Lily: "But I know my number!"
• Kate: "Now I know my number, too!"
• Jane: "I still don't know my number."

What is the sum of the numbers written on Kate and Lily’s hats?