Due to curiosity and also since I evaluated lower degree sums like these but this one is too hard to manipulate I am eager to know does this have a closed form ?
I broke it into the series $\displaystyle \sum_{m,n\ge 1}(-1)^{m+n}\frac{{\rm H}_m{\rm H}_n}{(m+1)(n+1)} \frac{2}{(2n-1)^3} $ , but does this help ?