Jointly Gaussian random vectors

It is not possible to write such a thing without knowing the covariance between the components of X and Y, or among different components of X and Y each among themselves.

If you do know that information, then simply break down X and Y in to scalar components, and write a jointly Gaussian distribution using a larger covariance matrix, which is a square matrix with the same number of dimension as the total number of scalar component. It really doesn't matter which one is part of $X$ and which one is part of $Y$, they are just a bunch of scalar random variables.

You state "We then have the joint density" etc. But that neglects the case where $K$ is singular. However, that is not essential to the question.

Perhaps normality is not essential to the question either.

If $X,Y$ are random variables that take values is $\mathbb R^m$ and $\mathbb R^n$ respectively, then $$ \left[ \begin{array}{c} X \\ Y \end{array} \right] \in \mathbb R^{m+n} $$ and one can write \begin{align} K & = \operatorname E\left( \left(\left[ \begin{array}{c} X \\ Y \end{array} \right] - \left[ \begin{array}{c} \mu_X \\ \mu_Y \end{array} \right] \right)\left( \left[ \begin{array}{cc} X^\top, & Y^\top \end{array} \right] - \left[ \begin{array}{cc} \mu_X^\top, & \mu_Y^\top \end{array} \right] \right) \right) \\[12pt] & = \left[ \begin{array}{cc} \operatorname E((X-\mu_X)(X-\mu_X)^\top & \operatorname E((X-\mu_X)(Y-\mu_Y)^\top) \\ \operatorname E((Y-\mu_Y)(X-\mu_X)^\top) & \operatorname E((Y-\mu_Y)(Y-\mu_Y)^\top) \end{array} \right] \\[10pt] & \in \mathbb R^{(m+n)\times(m+n)}. \end{align}

One also writes $$ \operatorname{cov}(X,Y) = \operatorname E((X-\mu_X)(Y-\mu_Y)^\top) \in \mathbb R^{m\times n} $$ and then one has $$ \operatorname{cov}(X,Y) = \big( \operatorname{cov}(Y,X)\big)^\top, $$ i.e., unlike in the scalar-valued case, the covariances with the arguments interchanged are not equal to each other, but are transposes of each other.

It will be a multivariate normal distribution.

Please refer to https://en.wikipedia.org/wiki/Multivariate_normal_distribution