Before you can start the Anna's Diary sidequest, you need to first complete the Where's the Dress! side quest, which is detailed here:
This side quest can be done during chapters 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, and 18.
After completing the Where's the Dress! side quest, York will find a key that falls out of the dress whilst Sallie is flinging it around. Go back upstairs into Anna's room with the key and read her diary which is on her desk.