I was able to compute the first part, with some computations I indeed got to:
$$\displaystyle{\sum_{i, k, h, j}} \delta_{ik}\delta_{hj}e_s(R_{hijk}) = n e_s(\lambda)$$
but for the other parts, I got all the way up to:
$$\displaystyle{\sum_{i, k, h, j}} \delta_{ik}\delta_{hj}e_j(R_{hiks}) = - \displaystyle{\sum_{j,h} \delta_{hj}e_j(\lambda \cdot \delta_{sh})}$$ $$\displaystyle{\sum_{i, k, h, j}}\delta_{ik}\delta_{hj} e_k (R_{hisj}) = - \displaystyle{\sum_{i,k} \delta_{ik} e_k(\lambda \cdot \delta_{is})}$$
and then got stuck. How are these both equal to $-e_s(\lambda)$? I'd appreciate some help.
You did all the hard work. Note, for example, that $$\sum_{j,h} \delta_{hj}\delta_{sh} e_j(\lambda) = \sum_j \delta_{sj}e_j(\lambda) = e_s(\lambda).$$