  We study which closed, connected, orientable three-manifolds $X$ containing a
Klein bottle arise as integral Dehn surgery along a knot in $S^3$. Such $X$ are
presentable as a gluing of the twisted $I$-bundle over the Klein bottle to a
knot manifold, and we use a variety of Heegaard Floer type invariants to
generate surgery obstructions. Suppose that $X$ is $8$-surgery along a genus
two knot, and arises by gluing the twisted $I$-bundle over the Klein bottle to
an $S^3$ knot complement. We show that $X$ is an L-space, it must be the
dihedral manifold $\left(-1; \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{2}{5}\right)$,
and the surgery knot must be $K=T(2,5)$.
