  In 1930, Kuratowski showed that $K_{3,3}$ and $K_5$ are the only two
minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of
the set of forbidden minors for any surface. \v{S}ir\'{a}\v{n} and Kochol
showed that there are infinitely many $k$-crossing-critical graphs for any
$k\ge 2$, even if restricted to simple $3$-connected graphs. Recently,
$2$-crossing-critical graphs have been completely characterized by Bokal,
Oporowski, Richter, and Salazar. We present a simplified description of large
2-crossing-critical graphs and use this simplification to count Hamiltonian
cycles in such graphs. We generalize this approach to an algorithm counting
Hamiltonian cycles in all 2-tiled graphs, thus extending the results of
Bodro\v{z}a-Panti\'c, Kwong, Doroslova\v{c}ki, and Panti\'c for $n = 2$.
