Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given Casimir Operator?
I only know how to find the quadratic one, using a physical analogy with the square of the 4-momentum, or the angular momentum.
After clarification: you are not searching for the Casimir element, you are looking for the generators of the center. Then you will need Harish-Chandra isomorphism So your problem will be equivalent to finding a basis in the ring of "$W$-symmetric'' polynomials $S(\mathfrak{h})^W$'. This is a subject of invariant theory. As $W$ is finite the problem is not to hard, look for Noether finiteness theorem (e.g. in Craft-Procesi Invariant theory: A primer). Of course finding reasonable basis is a problem that depends on properties you want it to have. For the $A,B,C,D$ series you can find the disscussion in Humphreys "Reflection groups and Coxeter groups'' Section 3.12
P.S. The term "Casimir elements" for the generators is popular in physics, but never used in mathematics.