How do we find force to derive $pV=Nmv^2$? 
Since momentum is not changing in time t , but instantaneously in very small time at wall B.
Why do we write $F=\frac{2mv}{t}$? Where $t$ is (twice the distance AB)/ velocity.
Why don't we take the instantaneous time to write the expression of force?
 A: I try to answer, not to solve the problem:
If you took an instantaneous time, you would get a quasi-infinite force and multiply it with very large number of particles, which is not going to help. And you get stuck with the mechanism of the recoil.
You should not care much about a single particle, rather care about what do all the particles and extract some macroscopic feature ... like $F=\sum_i \Delta p_i/\Delta t_i$ - one particle adds to the sum with the momentum transfered every x miliseconds and the details are not important in a large statistics. 
If you wanted to $\lim_{\Delta t \rightarrow 0}$, you would end up again with one recoil dynamics, but what you really care about is an average behavior.
A: 
I want to emphasize the fact that this is a very rough derivation and you can do more rigorous work using statistical mechanics. The pictorial representation i set up at the side is my interpretation based on how I learnt it and I am more than willing to correct it if it is wrong. Generally though, i think it's fine.
