Alright, the negative information matrix for $L(\mu,K)$ is

$$\frac{\partial^2 L}{\partial\mu\partial\mu} = -\left(\frac{1}{N}K\right)^{-1}$$

$$\frac{\partial^2 L}{\partial K\partial \mu} = 0$$

$\frac{\partial^2 L}{\partial K\partial K}$, in a more general formulation, is given at http://en.wikipedia.org/wiki/Fisher_information#Multivariate_normal_distribution

Thanks to Mike Hunter for finding it.

Alright, the negative information matrix for $L(\mu,K)$ is

$$\frac{\partial^2 L}{\partial\mu\,\partial\mu'} = -\left(\frac{1}{N}K\right)^{-1}$$

$$\frac{\partial^2 L}{\partial K\partial \mu} = 0$$

$\frac{\partial^2 L}{\partial K\, \partial K'}$, in a more general formulation, is given at http://en.wikipedia.org/wiki/Fisher_information#Multivariate_normal_distribution

Thanks to Mike Hunter for finding it.