As the battery remains connected, potential difference across the plate will remain the same.

Here, $C,U,V$ and $C^{\text{'}},U^{\text{'}},V^{\text{'}}$ are  capacitance, energy density of electric field and potential difference between the plates of the capacitor before insertion and after insertion of a slab. 

As the slab is introduced, 

$C^{\text{'}}=\left(k\right)\left(C\right)$

$U=\frac{1}{2}C{V}^2$

$V^{\text{'}}=V$

$U^{\text{'}}=\frac{1}{2}\left(C^{\text{'}}\right){\left(V^{\text{'}}\right)}^2$

$U^{\text{'}}=\frac{1}{2}\left(kC\right)(V{)}^2$

$U^{\text{'}}=k\left(U\right)$

Energy density increases $k$ times.