Start with this paper: A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9–32 (look also here). This reduces your problem to the case when $\Gamma$ is a lattice in $N\rtimes K$. Next, observe that projection of $\Gamma$ to $K$ has to be a finite group (this should be in M.Raghunathan, "Discrete subgroups of Lie groups"), the result is due to Auslender, you can read his original paper here. Now, use Jordan-Schur Theorem: For every compact group $K$ there exists a number $j=j(K)$ so that every finite subgroup of $K$ contains an abelian subgroup of index $\le j$. This is also in Raghunathan's book, see also wikipedia article here and Tao's blog here.