Aww. ;_; I felt pretty good about my deduction, too. ;_;
Okay, so the money was still there - which means that even if the aunt was the culprit, she wasn't motivated by the money. Then what was the motive? The printed note is funny - "You have done as I asked, and so your child shall return to you" - it sounds as if the culprit wanted the child, and not the money, and wanted to test the mother's love for the child. The culprit was probably upset that the mother worked 12 hours a day and didn't pay as much attention as she could have to the child . . . from this, it sounds like the housekeeper/babysitter was the culprit. Shinichi Natake says that the culprit's past is key to the motive - maybe the babysitter lost a boy Kurasuma-kun's age in the past? *Gasp* This is speculation, but maybe the husband has something to do with the housekeeper's child's death? Could this be the reason the husband died? Because the housekeeper killed the husband out of anger for the death of his child?
Anyways, how did the culprit get the child to go outside from the window? The housekeeper/babysitter probably told the child that they were going to play hide-and-seek sometime between 3:15 and 3:54. The housekeeper wanted to make it look like a kidnap, so he gave the child candy from his apartment and told the child to grab the mother's phone and a recording of the blackmail, and told the child to call home with the mother's phone and play the recording at the time when the play-date was supposed to happen. Since he is the housekeeper, he would have known about the time of the playdate. Then the housekeeper told the child to go outside through the window, maybe by telling the child that he should keep his hiding place as secret as possible. Then, when the mother left the money at the park, the housekeeper knew then that the mother loved the child deeply, so he called the child through the mother's cell phone and told the child to come back home. The housekeeper probably called from some phone booth as he was returning home, so that no one would recognize the phone number.
So in short, motive = to test the mother's love for the child, culprit = the housekeeper/babysitter, and trick = the child called home with a recording as part of what he thought was a game.
EDIT: Wait, the recording thing makes no sense - the child brought back only the cell phone, no recording. Maybe the housekeeper told the child to stash the recording after the child played it?