  We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$
at the prime 2, where $L_{K(n)}$ is localization with respect to Morava
$K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In
$L_1L_{K(2)}S^0$ we find all the summands predicted by the Chromatic Splitting
Conjecture, but we find some extra summands as well. An essential ingredient in
our approach is the analysis of the continuous group cohomology
$H^\ast_c(\mathbb{G}_2,E_0)$ where $\mathbb{G}_2$ is the Morava stabilizer
group and $E_0 = \mathbb{W}[[u_1]]$ is the ring of functions on the height $2$
Lubin-Tate space. We show that the inclusion of the constants $\mathbb{W} \to
E_0$ induces an isomorphism on group cohomology, a radical simplification.
