To get this off the "unanswered" list, there is the formula (which I also used here):
$\mathbf{A}^{-1}=\begin{pmatrix}\mathbf{E}^{-1}+\left(\mathbf{E}^{-1}\mathbf{F}\right)(\mathbf{H}-\mathbf{G}\mathbf{E}^{-1}\mathbf{F})^{-1}\left(\mathbf{G}\mathbf{E}^{-1}\right)&-\left(\mathbf{E}^{-1}\mathbf{F}\right)(\mathbf{H}-\mathbf{G}\mathbf{E}^{-1}\mathbf{F})^{-1}\\-(\mathbf{H}-\mathbf{G}\mathbf{E}^{-1}\mathbf{F})^{-1}\left(\mathbf{G}\mathbf{E}^{-1}\right)&(\mathbf{H}-\mathbf{G}\mathbf{E}^{-1}\mathbf{F})^{-1}\end{pmatrix}$
for the block matrix $\mathbf{A}=\begin{pmatrix}\mathbf{E}&\mathbf{F}\\ \mathbf{G}&\mathbf{H}\end{pmatrix}$.
Applying this formula to your matrix yields
$\begin{pmatrix}\mathbf{K}^{-1}-\left(\mathbf{K}^{-1}\mathbf{P}\right)(\mathbf{P}^\top\mathbf{K}^{-1}\mathbf{P})^{-1}\left(\mathbf{P}^\top\mathbf{K}^{-1}\right)&\left(\mathbf{K}^{-1}\mathbf{P}\right)(\mathbf{P}^\top\mathbf{K}^{-1}\mathbf{P})^{-1}\\(\mathbf{P}^\top\mathbf{K}^{-1}\mathbf{P})^{-1}\left(\mathbf{P}^\top\mathbf{K}^{-1}\right)&-(\mathbf{P}^\top\mathbf{K}^{-1}\mathbf{P})^{-1}\end{pmatrix}$