How does the mixing of two gases increase entropy?
Here is a reversible process to do what you are asking for (i.e., unmixing the gases reversibly).  You have a cylinder containing the two gases already mixed.  The cylinder has a piston.  Connected to this main cylinder are two other cylinders joined to it through semipermeable membranes.  Each membrane allows one of the gases to pass, but not the other.  You gradually advance the piston in the main cylinder while, at the same time moving the other pistons outward in the other two cylinders so that, at any time, the pressures of the pure gases in the attached cylinders are equal to their partial pressures in the main cylinder.  The change in entropy for this adiabatic reversible process is zero.  This is the first step.  Now, in the second step, we close off the semipermeable membranes, and compress each of the pure gases in the two cylinders isothermally and reversibly to the original total pressure that was present in the main chamber.  So, if $p_1$ was the partial pressure of one of the gases in the main chamber and $p_2$ was the partial pressure of the other gas in the main chamber, the total pressure of the pure gases in their chambers after completing step 2 will be $p_1+p_2$.  The change in entropy for the 2-step process we described is minus the entropy of mixing.  So, if the process is done in reverse, the entropy of mixing will be positive.