You have to turn into a cat and go back to the first bridge to like a stick thing that leads up to the other side. That's honestly the best way I can explain it. But if you just look beside the first bridge there is a stick thing leading up and just go on it in cat form.

In the room where the typewriter is, fly in the broom up toward the cieling. There will be two doors there. One of them leads to the broom race.

Where do you go to play chess?