(too lazy for emotes)

So, the poem...My broken heart, my broken wings. The Black left wing was found at the casino, where the 'wicked and greedy play,' so I suspect that the OTHER pieces of this puzzle will be found in the other locations. Coincidentally, the lockbox had a broken heart on it...

So, my suspicions on where the pieces are.

Scattered where the Violet Grapes used to grow-an abandoned Orchid. Someone search 37b, that used to be a Vineyard.
And forgotten where the children used to cry-an abandoned...Place. Detti's Nursery (40) seems to be the best bet.
Yearning to be United where the wind touches the waters-the lake, natch.

As for the white right wing...Resting where the Placid Orchids stay, eh? Dami's Flowerbed (13) is the most obvious choice.

Search those locations and head back to the appropriate locations. Put the heart pieces in the lockbox at the late (50), then head back to the apartments to unlock the door.
@Noblejanobii: Apparently any fully-evolved Pokemon could go do it and get a bonus. I suggest one with no type advantages/disadvantages with the Snorlax.