Is there any computer software available for solving for mixed strategy Nash equilibria for two players given each player's payoff matrix?
Yes. Here are two that I have co-authored:
The first one is succeeded by the second, which also solves two-player extensive-form games and offers more algorithms.
Both allow enumeration of all equilibria (including all equilibrium components for degenerate games) of bimatrix games, and only require one matrix to be input for zero-sum and symmetric games.
We welcome feedback on these, as they, in particular the second one, are in ongoing development.
For offline use, you should also check out the nash binary from the lrs program (which the web software above use as one of their algorithms):
For the underlying theory see e.g.:
D. Avis, G. Rosenberg, R. Savani , and B. von Stengel (2010). Enumeration of Nash Equilibria for Two-Player Games. Economic Theory 42, 9-37.
You should also be aware of the Gambit software package:
The web solvers above are distinct from but affiliated with the main Gambit software.
If you have Mathematica, there is a brute-force algorithm I authored. It is not efficient as the ones listed by Rahul Savani and it works well only for small games (say 4x4 or smaller). The only advantage is that if you are familiar with Mathematica you do not have to incur the cost of learning something new or install a system like Gambit.