sequences and series - Evaluating Sum involving binomial coefficients and powers

I would like to evaluate the double sum $\sum\limits_{n=1}^{\infty} \sum\limits_{m=1}^{\infty} \dfrac{(n+m)!}{n!m!n^2 m^2}\left(\dfrac{1}{2}\right)^{n+m}$. My starting point was to consider $\sum\limits_{n=1}^{\infty} \sum\limits_{m=1}^{\infty} \dfrac{(n+m)!}{n!m!}x^n y^m = \dfrac{1}{1 -x -y} - \dfrac{1}{1-x} - \dfrac{1}{1-y} + 1$ $\;\;\forall\;\; |x|+|y|<1\;\;$ All what is left is to divide by $xy$ then integrate with respect to $x$ and then with respect to $y$ (process should be repeated twice) finally set $x = y = \frac{1}{2}$. I am however stuck in evaluating the resulting integrals. I expect logarithmic and polylogarithmic functions to show up in the final result. I would appreciate if you can help me formulating the value of this sum.
Thanks for your help...