$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$
where,
$n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ $\sum_{h=1}^{L}N_h=N$
My Attempt :
$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}$$ $$=\sum_{h=1}^{L}\frac{W_h^2S_h^2}{\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h}$$ $$=\sum_{h=1}^{L}\frac{W_h^2S_h}{\frac{n}{\sum_{h=1}^{L}N_hS_h}N_h}$$ $$=\sum_{h=1}^{L}\frac{W_h^2({\sum_{h=1}^{L}N_hS_h})S_h}{nN_h}$$
You have $$W_h={N_h\over N}$$ but $N$ doesn't appear anywhere else in the question, so I'm going to assume that's a typo for $$W_h={N_h\over n}$$ Then $$N_h=nW_h$$ and $$n_h={nN_hS_h\over\sum N_jS_j}={n^2W_hS_h\over\sum nW_jS_j}={nW_hS_h\over\sum W_jS_j}$$ so
$$\sum{W_h^2S_h^2\over n_h}={\sum W_hS_h\sum W_jS_j\over n}={1\over n}\left(\sum W_hS_h\right)^2$$ which isn't exactly what you wanted, but I think it's correct.
EDIT: It was correct, until OP edited another condition ($\sum N_h=N$) into the question. I'm happy to let someone else take the risk that OP has now settled on the final form of the question.