Headlong; at full speed.
This is an American phrase in origin, possibly with Scottish influences, and isn't commonly used in other countries. Lickety may be taken from lick, meaning speed - as in 'going at quite a lick'. That usage is known by the early 19th century; for example, this piece from Thomas Donaldson's Poems, chiefly in the Scottish dialect, 1809:
"Ere I get a pick, In comes young Nannie wi' a lick."
It is variously spelled in early citations but, whatever the spelling, it is just as likely to be a nonsense word, not pertaining to anything in particular. The first record of it in print is in D. McKillop's Poems, 1817:
"I rattl'd owre the A, B, C, as fast as lickitie An' read like hickitie."
The hiciktie in that line may be a version of heck - itself a euphemism for hell. I can't find out anything about Mr. McKillop but I would guess he was a Scottish gentleman - Donaldson certainly was. Lickitie in that spelling certainly wouldn't look out of place in Scotland.
The second word of the term is just an intensifier, and 'split' was settled on eventually. That is first cited in American Speech, 1848, as 'lickoty split'. Lickety may have been imported into the USA via immigration from Scotland. Split seems to have been added in the USA.
The many variations on 'lickety split', for example 'licketty cut', 'lickety click', 'lickoty split' suggest an invented onomatopoeic phrase. It is suggestive of phrases like clickety-click which mimic trains running across points.
Also of American origin is the more recent vulgar usage of the term to mean cunnilingus. This isn't common even in the USA and dates from the 1960s. It first appeared in print in the jokes section of Playboy Magazine, January 1970, in a joke about Mae West which I'll leave to your imagination.