Revealing mistake: Kate's birthday is July 10th. When she is (with long hair) outside on the patio with Alex, you can see their breath as they talk, and there is not a living thing in the back yard, it's all dormant (as in late fall/winter).
Revealing mistake: When Kate kisses Morgan in front of 1620 N. Racine, the camera moves up through the branches of a tree. You can see one of the lower branches shake when it is bumped by the crane. (01:17:20)
Answer: In her apartment in Chicago. Because Alex knew her address, he hid it in the floorboard for her to eventually find. He knew how important the book was to her. He didn't want to send it to her in the mail, as she originally requested.