Using 3D cylindrical coordinates, I get 0 as the answer.

$$ \nabla^{2} (k \hat{r}) = \hat{r} (\frac{1}{r} \frac{\partial }{\partial r}\left(r \frac{\partial (k)}{\partial r}\right) + 0 + 0) + \hat{\phi} [\nabla^{2}(0)] + \hat{z} [\nabla^{2}(0)]= 0$$

Am I correct?

The solution to this question in the book that I'm using has ignored the unit vector and has given the Laplacian of $kr$ instead which is $k/r$.