If a matrix is differentiated with respect to itself, the result should be a fourth order tensor. The easist way to see this is to work with components.
$$\frac{ \partial K_{ij}}{\partial {K_{kl}}} = \delta_{ik}\delta_{jl}$$
$\delta_{ij}$ is the Kronecker delta. Because differentiation survives only when $i=k$ and $j=k$. If your matrix is symmetric, then
$$\frac{ \partial K_{ij}}{\partial {K_{kl}}} = \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}$$
Because, the differentiation also survives for $i=l$ and $j=k$.