kernel to calculate second order partial derivative of digital image
Your five-point derivative kernel is a 1D kernel. Applied along the x axis it gives the partial derivative for x, applied along the y axis it gives the partial derivative for y.
The $\frac{\partial^2}{\partial x \partial y}$ derivative would need a 2D square kernel. It is more efficient to compute this by applying two first order partial derivatives.
$$\frac{\partial^2}{\partial x \partial y} f = \left( \frac{\partial}{\partial x} \frac{\partial}{\partial x} \right) f = \frac{\partial}{\partial x} \left(  \frac{\partial}{\partial x} f \right)$$
Since $\frac{\partial}{\partial x} f$ is a convolution,
$$\frac{\partial}{\partial x} f = k_x \ast f$$
we can write the earlier equation as
$$\frac{\partial^2}{\partial x \partial y} f = \left( k_x \ast k_y \right) \ast f = k_x \ast \left(  k_y \ast f \right)$$
So we can say that the 2D square convolution kernel that you are looking for is $k_x \ast k_y$ (with $k_x$ the kernel that computes the x-derivative and $k_y$ the one that computes the y-derivative). Note that, if you think of your 1D kernels as vectors, this convolution is equivalent to the outer product of the two vectors.
However, it is more expensive to compute a convolution with one 5x5 kernel (25 multiplication and additions per pixel) than to compute two convolutions with 5x1 and 1x5 kernels (10 multiplications and additions per pixel). I would compute two first order derivatives in sequence.