  We consider double scaled little string theory on $K3$. These theories are
labelled by a positive integer $k \ge 2$ and an $ADE$ root lattice with Coxeter
number $k$. We count BPS fundamental string states in the holographic dual of
this theory using the superconformal field theory $K3 \times \left(
\frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/
\mathbb{Z}_k$. We show that the BPS fundamental string states that are counted
by the second helicity supertrace of this theory give rise to weight two mixed
mock modular forms. We compute the helicity supertraces using two separate
techniques: a path integral analysis that leads to a modular invariant but
non-holomorphic answer, and a Hamiltonian analysis of the contribution from
discrete states which leads to a holomorphic but not modular invariant answer.
From a mathematical point of view the Hamiltonian analysis leads to a mixed
mock modular form while the path integral gives the completion of this mixed
mock modular form. We also compare these weight two mixed mock modular forms to
those that appear in instances of Umbral Moonshine labelled by Niemeier root
lattices $X$ that are powers of $ADE$ root lattices and find that they are
equal up to a constant factor that we determine. In the course of the analysis
we encounter an interesting generalization of Appell-Lerch sums and
generalizations of the Riemann relations of Jacobi theta functions that they
obey.
