What is the value of the gas constant derived from the kinetic model of gases?
$T$ here does not refer to temperature but rather the average kinetic energy of a molecule.

I can see why there is some confusion. Sadly we only have so many symbols and kinetic energy also happens to be commonly denoted with $T$.

To relate the average kinetic energy to the temperature, you need to bring in equipartition theory

$$\frac{1}{2}m\langle v^2\rangle = \frac{3}{2}kT$$

(assuming three translational degrees of freedom only). If you substitute that into your above equation

$$pV = \frac{2}{3}nN_\mathrm{A}\cdot m\langle v^2\rangle$$

then you will find that $pV = n(N_\mathrm{A}k_\mathrm{B})T \Longrightarrow R = N_\mathrm{A}k_\mathrm{B}$, hopefully a familiar expression. The question of how the value of $R$ is derived then becomes a question of how equipartition is derived - and that deserves another question itself.