Green’s Dictionary of Slang

like sixty adv.

also sixty
[? SE like sixty men; on model of like twenty adv. and like forty adv.]

[mid-19C+] (orig. US) with great force or vigour, at a great speed.

In phrases

swing like sixty (v.) [swing v. (9a) ]

[1960s+] (US teen) to perform at one’s peak, to achieve ultimate success or pleasure.