This problem was studied by J.E. Mazo and A.M. Odlyzko in <A HREF="http://www.dtc.umn.edu/~odlyzko/doc/arch/high.dim.spheres.pdf">Lattice Points in High-Dimensional Spheres</A>. If $n$ grows with $k$ faster than $\sqrt k$ then the volume asymptotic still applies.