The song is a poem written by Eugene Field and first published in 1892. Heather only sings the first verse but the full poem has three verses that each highlight a different figure: the storm king in the first, the mother pine in the second, and the singer (presumably the mother who is singing to her child).

Here is the full poem:

"The sky is dark and the hills are white As the storm-king speeds from the north to-night, And this is the song the storm-king sings, As over the world his cloak he flings:  "Sleep, sleep, little one, sleep;" He rustles his wings and gruffly sings:  "Sleep, little one, sleep."

On yonder mountain-side a vine Clings at the foot of a mother pine; The tree bends over the trembling thing, And only the vine can hear her sing:  "Sleep, sleep, little one, sleep; What shall you fear when I am here?  Sleep, little one, sleep."

The king may sing in his bitter flight, The tree may croon to the vine to-night, But the little snowflake at my breast Liketh the song I sing the best,—  Sleep, sleep, little one, sleep; Weary thou art, anext my heart  Sleep, little one, sleep."

This website goes deep into the meaning of it if you're interested.

The main point of my research was actually to find sheet music for it and I discovered that four or five people have composed different music for the poem: David Barton, Ralph L. Baldwin, Norman O'Neill, Anna Marie Maddalena Gonzalez, and Hannah Brown (this one is on Musescore so it's probably not very official but I really like this one so I felt the need to at least mention this composer/score). Most of the scores cost money :C so I couldn't play them but the one on Musescore and the one by Baldwin are free use: