The below identity I have found experimentally.

Question. Is this true? If so, may you provide a "slick" (or any) proof. $$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}\frac{kq^k}{1-q^k}\right)^2=\sum_{k=1}^{\infty}\frac{(5k^3+k)q^k}{1-q^k}.$$