sixtieth (adj., n.)

"one more than fifty-nine, twice thirty, six times ten; the number which is one more than fifty-nine; a symbol representing this number;" Middle English sixti, from Old English sixtig, from siex (see six) + -tig (see -ty (1)). Similar formation in Old Norse sextugr, sextögr, sextigir, Old Frisian sextich, Middle Dutch sestig, Dutch zestig, Old High German sehszug, German sechzig.

To do something like sixty "with great force or vigor" is by 1833, American English, but the signification of the sixty is unclear.

 "... you know she paints admirably, (all I know is that she daubs like sixty, and calls it painting) ..." [North Carolina Sentinel, Newbern, N.C., July 19, 1833, reprinted "From the Auburn Journal," written in the voice of a common man]

The variant like sixty-six is more recent (by 1853 in North Carolina newspapers) and thus might be a mere embiggening of it.

word-forming element making ordinal numbers (fourth, tenth, etc.), Old English -ða, from Proto-Germanic *-tha- (cognates: Gothic -da, -ta, Old High German -do, -to, Old Norse -di, -ti), from PIE *-to-, also *-eto-, *-oto-, suffix forming adjectives "marking the accomplishment of the notion of the base" [Watkins].

Cognate with Sanskrit thah, Greek -tos, Latin -tus; Sanskrit ta-, Lithuanian and Old Church Slavonic to, Greek to "the," Latin talis "such;" Greek tēlikos "so old, of such an age," Old Church Slavonic toli "so, to such a degree," toliku "so much," Russian toliko "only;" also see -ed.