The Johnson solids are the strictly convex polyhedra made out of regular faces, excluding the vertex-transitive ones (which instead enter into the uniform category). Victor Zalgaller proved in 1969, in his paper Convex Polyhedra With Regular Faces, that only $92$ such polyhedra existed. I've read that he did this by a tedious computer search. I’m not aware of any other document that covers this proof.
The problem is, even after scouring the internet (and even some not fully legitimate sites) for about an hour, I've been unable to find a copy of his paper anywhere (except for this one, in Russian). And I have no idea of how I would replicate the argument myself. It doesn't even seem trivial that there is a way to reduce the problem to finitely many calculations. In fact, the existence of prisms and antiprisms seems to effectively contradict that intuition: what's stopping us from taping a bunch of equilateral triangles to a $50$-gonal prism and creating a valid solid, for example? [EDIT: Ted's answer already redirects to another source explaining this.]
I'm not asking for a full rundown of the proof, that would be too much to ask. My question is:
What are the main ideas of Zalgaller's (or any other's) enumeration of the Johnson solids?
Any accessible reference is welcome.