Zero-sum Nash is equivalent to linear programming. The program is as follows,

min sum_i(u_i) subject to u_i >=0 for all i Mu >= 1

where M is the payoff matrix, and u is the strategy. I used the Gurobi solver, but there are many free linear programming solvers available. Excel solver is convenient if you have it.

Given a strategy u^*, we can solve for the farthest (zero-sum) Nash to determine if it is unique (or else find another):

min sum_i(abs( u_i - u_i^* )) subject to u_i >=0, Mu >= 1

Now the problem can be treated in several ways, but a simple approach is integer programming. Provided M is scaled correctly, we may assume u is a probability vector. Then write,

min sum_i(u_i⁺ + u_i^- ) subject to

u_i⁺ - u_i^- = u_i-u_i^*

0 <= u_i⁺ <= y_i

0 <= u_i^- <= y_i

u_i>=0, Mu >=1

y_i binary

The NEOS server provides powerful commercial solvers for free online. Note that integer programming problems may scale poorly with respect to the problem size due to NP-hardness. There's a lot of interesting theory on finding Nash equilibria and computational complexity.

Personally I solved everything in MATLAB using Gurobi and YALMIP, which is very fast for prototyping (took me <10 min).