Kirkpatrick chose the 60 sonatas for analytical reasons to accompany his landmark 1953 book on Scarlatti, and later handed them off to Schirmer for publication. There are complete bound editions out there (I think Kalmus has one) but they're also available for download at
https://imslp.org/wiki/Category:Scarlatti,_Domenico; IMSLP has both the Longo edition, and also a couple of hundred sonatas edited by the harpsichordist John Sankey.
I don't believe any of the existing manuscripts were actually written by Scarlatti himself; they're all copies or copies of copies, and some of the copying appears to have been done by people of dubious musical knowledge, so you'll find any number of variants.
The catalog numbering by Kirkpatrick is not "off"; it differs from Longo because part of what made Kirkpatrick's work so pioneering was that he was the first person to systematically attempt to put these works in their true chronological order (and also support his contention that the sonatas were composed in pairs).