For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is finite or not.

I am not sure of your notation but if you want the L² dimension of the regular representation l²G of a discrete group G, then it is 1 more or less by definition. In fact, one can think to L²dimension of a representation of G as a dimension relatively to l²G. 

