Before collision :

After collision :

We can resolve the momentum in a direction along the line of impact and perpendicular to it.
Velocity perpendicular to the line of impact will not change. Balls are identical and collision is elastic, so velocities get interchanged along the line of imapct.

$\therefore {\mathrm{v}}_1=0,{\mathrm{v}}_2=\frac{\mathrm{u}}{2}=\frac{\mathrm{u}}{\mathrm{k}}\Rightarrow \mathrm{K}=2$