Answer
(a) $V = 10 \,\text{V}$
(b) $K = 2.25$
Work Step by Step
(a) The battery is kept connected which means the potential is constant in both states and we will use the value of the energy stored to get the potential
\begin{gather*}
U = \dfrac{1}{2} C V^2 \\
V =\sqrt{\dfrac{2 U}{C}} \\ \
V = \sqrt{\dfrac{2\times 1.85 \times 10^{-5} \,\text{J}}{360 \times 10^{-9} \,\text{F}}}\\
\boxed{V = 10 \,\text{V}}
\end{gather*}
(b) The energy stored is directly proportional to the dielectric constant $K$ therefore, we could get $K$ by
$$K = \dfrac{\Delta U + U}{U} = \dfrac{ 2.32 \times 10^{-5} \,\text{J} + 1.85 \times 10^{-5} \,\text{J}}{ 1.85 \times 10^{-5} \,\text{J}} =\boxed{ 2.25} $$
