Green’s Dictionary of Slang

sleep; thus have a dose of the balmy v., to sleep; occas. ext. to death.

1841		Dickens Old Curiosity Shop (1999) 74: As it’s rather late, I’ll try and get a wink or two of the balmy.
1861		Bell’s Life in Sydney 2 Feb. 3/2: Her caro sposo, who was doing the balmy, [...] was aroused by the noise of her clapper.
1864, 1867, 1870		Hotten Sl. Dict.
1873		Sl. Dict.
1881		Newcastle Courant 2 Dec. 6/5: Hush-hush, he’s taking a dose of the balmy, but he skippers as lightly as a mouser.
1882		Sydney Sl. Dict. (2 edn) 1: Balmy - Sleep. To ‘take a dose of the balmy.’.
1888		H. King Savage London 386: Stow that, or I’ll give you a taste of the balmy which won’t be cured this side of kingdom come.
1890–1904		Farmer & Henley Sl. and Its Analogues I 110/2: To have a dose of the balmy; i.e., ‘to go to sleep’.