We have to this point glossed over the complications created by the book, in large part because what is most interesting about it is not that he acquires it but how he returns it.
We noted that Alex begins to attend Kate's 2004 birthday party in that revision of history in which he has the dog, even though at that point he does not recognize Kate as the girl who sent him the letter. With each letter his knowledge of her grows, and thus at some point he makes the connection, that the Kate whose birthday he is attending is the same girl who is writing to him from two years in the future. However, when she asks him to recover her book, he gets the piece he needs to connect with her at the party by asking her about the story. Thus the book shifts that event into what we see in the film.
Yet when he receives the book, he does not immediately return it to her. Rather, he tells her that he will do so. It may be that his plan was to hand it to her in person in the future which for him would never come; he never explains what his original intention was. However, it seems to have been derailed when she terminates their correspondence and refuses to read his additional letters.
She eventually finds the book in the floor of the bedroom of her apartment. This is a very odd scene. Undoubtedly she has been in that room almost every day since she left the lake house, and it has by now been over a year. (It must be the Riviera Condominiums apartment--there is no way Alex could have known where she would live after that.) Yet on this particular day she steps on the rug and realizes that the floor is loose beneath it.
This again recalls Frequency, when John tells his father to hide the wallet behind a loose board in the house and immediately retrieves it in the future. That makes sense, because if the wallet was put there in 1969 and never removed, it will still be there in 1999 for John to find.
The problem with the same trick in The Lake House is that the floor board has to have been loose. That is, if Alex entered the construction site in 2004 and hid the book under the floor, leaving a floorboard loose, then the floorboard was loose the day Alex moved into the apartment, and she probably discovered it long even before she asked him to recover it for her. Yet she finds it after most of a year there, simply because she stepped on the loose board. It is inconceivable that she would never have stepped on that board before that day; how could it not have been loose then but be loose now?
The answer would appear to be similar to the question of why the tree appears in front of her apartment: Kate didn't find the book which, like Frank's wallet, had been hidden in the floor for two years. In putting it in the floor, Alex sent it to her, just as he sent all the letters and the tree and the grafitti. It does not hide under a loose floorboard. It leaps forward two years to be hidden under a suddenly loosened floorboard in Kate's time.
This is very peculiar. It is one thing to say that the book, like the tree and the grafitti and the letters, leapt forward two years to appear in Kate's bedroom at that time. The problem is not so much the book as the floorboard. If Alex entered the nearly finished apartment, loosened a floorboard, and placed the book beneath it, one of two things would have happened: either someone would have fixed it, or they would not have done so. If they did fix it, they probably pulled it up and noticed the book beneath it, so the book would have been removed and the floor would never have been loose. If they did not fix it, then it would always have been loose, and she would have come upon the loose board (whether or not the book was beneath it) long before this.
That it is a magical fantasy can only explain so much; but perhaps this has to be written off to the magic. Whether Alex fixed the floor himself or left it undone, the magic delivered the loose floorboard along with the book two years after he hid it in her future apartment. It doesn't exactly break the rules of the story, but it stretches them a bit further than before, and it seems unlikely that Alex could have predicted that result.
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