The song can be found in Worship in the Family and School-Room: A Manual of Devotion by William Theodore Wylie (Springfield, Mass.: W. J. Holland & Co., 1867), page 104, under the title GLORY TO GOD. It is attributed to Rev. Alfred Taylor, and appears with musical notation for 4 voices. It has an additional verse (the 2nd) not in the Copper's version:

Day in the east is breaking—day o'er the crimsoned earth;
Now the glad world is waking, glad in the Saviour's birth!
See where the bright star bendeth over the manger blest:
See where the infant Jesus smiles upon Mary's breast!
Glory to God!—we hear again,
Peace on the earth, good will to men!

The other 3 verses agree exactly (except for punctuation) with those posted above.

However, I also found it in other later books with other titles, for example, PEACE ON EARTH and CHRISTMAS.

Maybe it's just as well to stick to the time-honored custom of calling hymns by their first line—SOFTLY THE NIGHT IS SLEEPING.