Finding the maximum possible size of S, where S is a set of pairwise-disjoint subsets of the list, and every element of S sums to k
This problem is NP-complete.  Even testing whether there exists a single subset that sums to $k$ is NP-complete; that is known as the subset sum problem.  Therefore, you should not expect any efficient solution to this problem.

If you are asking for any way to solve it, without regard to efficiency, one approach is to enumerate all possible subsets that sum to $k$, build a graph with an edge between each pair of subsets that are not disjoint, and search for a maximum independent set in this graph.  Another approach would be to use integer linear programming; define the zero-or-one $x_{ij}$ integer variable to be 1 if the $i$th subset includes the element $j$, and then add constraints to reflect the problem statement, i.e., $\sum_j x_{ij} j = k$ and $x_{ij} + x_{i'j} \le 1$ for all $i\ne i'$.  Then, apply an off-the-shelp ILP solver.