It appears as if the most commonly attributed author of the song is an A. D. Fitz Randolph.

The book The Blue Sky Boys by Dick Spottswood credits the poem to Samuel Irenaesmus Prime's Thoughts on the Death of Little Children and attributed it to Prime in the year 1852. This, however, is at least partially wrong: you can read (or, at least, attempt to read, due to poor preservation quality) a copy of the poem in the Monmouth Inquirer [Freehold, NJ] from Oct. 25, 1851.

The poem appeared in several newspapers throughout the 1850s, but unfortunately it was never given an attribution.

Despite this, several sources say the song was by Randolph. The Shadow on the Hearth (1858) and Fading Flowers (1860) both credit the song to Randolph. More credence is given to this claim by the publication of Randolph's book, Hopefully Waiting, and other poems in 1879, though it should be noted that Randolph's book also included "many other stray pieces collected by him" in his book, and made no effort to indicate which was which.

I've found a very long edition of the poem which contains nearly all the stanzas from the post of Stewie back in 2002 (albeit, in a very different order) except for the one I'll quote below. Half of its stanza matched one provided, while the other didn't (note that this copy had the stanzas as 8 lines instead of 4 lines each); since you need the context of the first 4 lines for the second, I decided to quote the whole thing.

All the day, while you were working,
As I lay upon my bed,
I was trying to be patient
And to think of all you said--
How the kind and blessed Jesus
Loves his lambs to watch and keep,
And I wished he'd come and take me
In his arms, that I might sleep.