Relation between R and k
The form of the equation is correct, but one of the variables is wrong. It should be:

$$\mathrm{R} = N_{\mathrm A}k_{\mathrm B}$$

The Boltzmann constant ($k_{\mathrm B}$) is for one molecule (or atom); the universal gas constant ($R$) is for one mole of molecules (or atoms). 

Therefore, in this relationship - and which is clear from dimensional analysis ($R$ is per mole) - $N_{\mathrm A}$ is the Avogadro constant, and not number of moles: in your argument, $n$ is a dimensionless integer, and thus the units do not work out. You need to use $N_{\mathrm A}$ instead of $n$. Note that the Avogadro constant can be denoted by either $L$ or $N_{\mathrm A}$.

The proper relationship is:

$$R = (\pu{6.022*10^{23} mol-1}) \cdot (\pu{1.380648*10^{-23} J*K-1}) = \pu{8.314 J*K-1*mol-1}$$

Note that I have included more significant digits for the Boltzmann constant here to get your value for $R$.