According to this article, page 673:
The "von Kármán line" appears to be what mathematicians refer to as a "folk theorem" , arising out of a conference discussion but never formally published by him. It was fleshed out in later publications, especially in the influential work of Haley(1963) and there is some justification for calling it the "von Kármán-Haley line".
The Theodore von Kármán collection is a guide to the original collection of papers, personal correspondence, scientific notes, illustrations etc. and there is also a microfiche publication of this collection.
The personal correspondence section of the collection in box 12 also has the correspondence with Andrew Haley and there is a good chance that in some of those letters calculations regarding the altitude of the Kármán line are present. And in some scientific notes as well of course.