  We study null Sasakian structures in dimension five. First, based on a result
due to Koll\'ar [Ko], we improve a result by Boyer, Galicki and Matzeu in [BGM]
and prove that simply connected manifolds diffeomorphic to $# k(S^2\times S^3)$
admit null Sasaki $\eta$-Einstein structures if and only if $k\in \{3,...,
21\}$. After this, we determine the moduli space of simply connected null
Sasaki $\eta$-Einstein structures. This is accomplished using information on
the moduli of lattice polarized K3 surfaces.
