This is a quintessential cradle song written for the piano, intended to evoke the calm of a nursery. It is musically simple, relying on rocking broken-chord figures and a soothing melodic contour.
Focus on achieving a perfectly even and hushed dynamic level throughout the performance.
Comparable difficulty and character — good candidates for what to learn next.
'Sleep! Baby, sleep!' is rated Henle level 2 of 9 — beginner repertoire, roughly ABRSM grades 2–3. In the RCM system it corresponds to about level 2.
You should be comfortable at Henle level 2 — meaning you can already play level 1–2 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of 'Sleep! Baby, sleep!' lasts about 2 minutes.
'Sleep! Baby, sleep!' is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Autumn song by Alfred Humphries Pease, 'My little love' by Alfred Humphries Pease, 'Tis lone on the waters by Alfred Humphries Pease — see the full similar-pieces list on this page.