  We consider type II superstring theory on $K3 \times S^1 \times
\mathbb{R}^{1,4}$ and study perturbative BPS states in the near-horizon
background of two Neveu-Schwarz fivebranes whose world-volume wraps the $K3
\times S^1$ factor. These states are counted by the spacetime helicity
supertrace $\chi_2(\tau)$ which we evaluate. We find a simple expression for
$\chi_2(\tau)$ in terms of the completion of the mock modular form
$H^{(2)}(\tau)$ that has appeared recently in studies of the decomposition of
the elliptic genus of K3 surfaces into characters of the N=4 superconformal
algebra and which manifests a moonshine connection to the Mathieu group
$M_{24}$.
