Green’s Dictionary of Slang

(US drugs) the Federal Narcotics Hospital, Lexington, Kentucky.

1964		Larner & Tefferteller Addict in the Street (1966) 207: Roy had kicked at K-Y [Lexington, Kentucky] but he started in again.
1966		J. Mills Panic in Needle Park (1971) 75: After a while I got really strung out again and I went to KY. And I was there maybe a month.
1973		M. Agar Ripping and Running 138: He’s this dude we did the bit with down at K-Y.
1989		Courtwright & Des Jarlais Addicts Who Survived 298: Among addicts it was referred to variously as ‘Narco,’ ‘Lex,’ ‘Lexington,’ or simply ‘K.Y.’.