Here is the whole song, with what it seems is a good translation (per Monique and Q and other sources).    Some phrases can probably be worded more simply in English without losing the meaning (e.g., "Let Him be blessed" is pretty much the same thing as "Bless Him" and we might find a better English word for "delirio" than either "delirium" or "frenzy") but those little variations probably don't matter.   A "singable translation" would be another matter, and some songs don't lend themselves well to "translations" that rhyme and also scan with the cadence of the music.

A LA NANITA (LITTLE LULLABY)

Coro:
A la nanita, nana, nanita eya, nanita eya.
Mi jesus tiene sueño. Bendito sea, bendito sea.
(bis)

[Chorus:
Little lullaby, lullaby, little lullaby eya, little lullaby eya.
My Jesus is sleeping.
Let Him be blessed. Let Him be blessed.]

1. Fuentecita que corre clara y sonora,
Rui señor que en la selva cantando llora
Calla mientras la cuna se balansea
A la nanita nana, nanita eya. (Coro)

[Little spring that flows clear and sonorous,
Nightingale singing mournfully in the forest,
Keep it quiet while the cradle rocks
to this little lullaby. (Cho.) ]

2. Pimpollo de canela, lirio en capullo,
duérmete sin recelo, mientras te arrullo,
duérmete sue del alma mi canto brota,
y un delirio de amores es cada nota. (Coro)

[Cinnamon sprig, budding lily,
Sleep without fear while I lull you.
Sleep, as my song springs up from my soul
And every note is a frenzy of love. (Cho.) ]

3. Manojito de rosas y de alelíes,
?Que es lo que estás soñando, que te sonries?
?Cuáles son tus ensueños? Dilo, alma mia;
Mas, ?Qué es lo que murmuras? Eucaristia?. (Coro)

[Little handful of roses and wallflowers,
What are you dreaming of, what makes you smile?
What are your dreams? Tell me, my soul.
And what are you whispering? The Eucharist?   (Cho.) ]

4. Yo no sé lo que es eso, Niño del alma,
mas pues esa sonrisa mis penas calma,
sigue, sigue soñando, mi dulce dueño,
sin que nada ahuyente tan dulce sueño. (Coro)

[I don't know what this is, Child of my soul,
But since this smile soothes my sorrows,
Dream on, dream on, my sweet Lord,
With nothing scaring away such a sweet dream. (Cho.) ]


5.   Pajaritos y fuentes, auras y brisas,
respetad ese sueño y esas sonrisas;
calla mientras la cuna se balancea,
El Niño está soñando, !Bendito sea! (Coro)

[Little birds and fountains, auras and breezes
Honor that dream and those smiles.
Hush while the cradle rocks,
The (Holy) Child is dreaming. Let Him be blessed.