Evaluate $\sum_{n\ge0}\sum_{m\ge0}\frac{\sin((2n+1)k)\sin((2m+1)k)\sinh(k\sqrt{(2n+1)^2+(2m+1)^2})}{(2n+1)(2m+1)\sinh(\pi\sqrt{(2n+1)^2+(2m+1)^2})}$

$$V=\sum_{n_x}\sum_{n_y}\frac{1}{n_xn_y}\frac{\sin(kn_x)\sin(kn_y)\sinh\left(k\sqrt{n_x^2+n_y^2}\right)}{\sinh(\pi\sqrt{n_x^2+n_y^2})}$$

Here, $n_x, n_y$ are odd integers and $k=\frac{\pi}{2}$

I obtained this sum while trying to solve for potential inside a cube.

I have been trying to evaluate the closed form without Mathematica, and there's not much progress or work I have to show for this problem in particular, as obtaining this was a long process in itself.

I attempted trying to follow along the methods described in this paper (was too complex)

Rewriting the sum as,

$$V=\sum_{n=0}^\infty\sum_{m=0}^\infty\frac{1}{(2n+1)(2m+1)}\frac{\sin((2n+1)k_1)\sin((2m+1)k_2)\sinh\left(k_3\sqrt{(2n+1)^2+(2m+1)^2}\right)}{\sinh(\pi\sqrt{(2n+1)^2+(2m+1)^2})}$$

I just have the information that this sum approximates to $\frac{\zeta{(2)}}{16}$

I'm interested only in methods in how to solve for this sum and similar kinds, without relying on any CAS.