  Square-tiled surfaces can be classified by their number of squares and their
cylinder diagrams (also called realizable separatrix diagrams). For the case of
$n$ squares and two cone points with angle $4 \pi$ each, we set up and
parametrize the classification into four diagrams. Our main result is to
provide formulae for enumeration of square-tiled surfaces in these four
diagrams, completing the detailed count for genus two. The formulae are in
terms of various well-studied arithmetic functions, enabling us to give
asymptotics for each diagram using a new calculation for additive convolutions
of divisor functions that was recently derived by the author and collaborators.
Interestingly, two of the four cylinder diagrams occur with asymptotic density
1/4, but the other diagrams occur with different (and irrational) densities.
