I have a hypothesis, but I'm not sure if its true.
The Wikipedia page states that if the covariance matrix is given by $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ the conditional covariance matrix is given by $$\operatorname{Cov}(X\mid Y) = A-BC^{-1}B^T$$
Slide 23 in this presentation on graphical models states that $$K_{11}^{-1}=\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$$ which is the same as the above formula!
So it seems like there is an elegant way to find the covariance of $X|Y$. Simply look at the corresponding block in the precision (inverse covariance) matrix and take its inverse. Is this true? If it is, it seems like such a cool result. Why can't I see it explicitly spelled out anywhere?