REGISTER TO REMOVE ADS
I remembered half of the method!You can't jump onto the balcony or the hedge, but there's one pathway in the Bizarre Room that leads to one of the hedges. I forget which one it is, though. Once you're on it, just like Taochan said, you can just glide your way onto the balcony, although you need to be careful not to fall down if you don't want to repeat all over again.
If you're curious as to what the chests contain, the one on the balcony has a Dark Matter. The other one... if not a trio of puppies, a fairly good gummi block.