Unfortunately the Kalmus edition is by Longo (responsible for the L numbering), who first brought the sonatas to light, but who also changed them here and there and added many spurious performance directions (now discredited).
Ralph Kirkpatrick (responsible for the K numbering) published a selection of 60 for Schirmer, and his edition is authoritative. He also published facsimiles of the original copies. Kenneth Gilbert then edited a urtext edition based on Kirkpatrick’s facsimiles and which is arguably the most authoritative. It is published by Heugel (Le Pupitre) and consists of 11 volumes. It is very expensive, but I am very lucky in that my local library has the whole collection. I have the CD Joel mentioned above, which is the Longo (unreliable) edition, so I always check the score with the Kenneth Gilbert one. Here is their site:
https://www.cdsheetmusic.com/ Henle is in the process of publishing the Urtext but as far as I know there are only selected sonatas available at the moment.
Finally, no one knows how many sonatas Scarlatti actually composed (there are no autographs, just copies), so no one can claim to have published the complete set, but the ones above are as complete as possible at the moment.
I hope this helps,
Best wishes,
Bernhard.