Let $\Omega \subset R^2$ is a ball. Consider the equation
\[
\begin{cases}
-\triangle u = f(x), \quad x \in \Omega \\
u \big|_{\partial \Omega} = 0.
\end{cases}
\]
It suffices to prove that for $p \geq 2$
\[
\|D^2u\|_{L^p(\Omega)} \leq C \|f\|_{L^p(\Omega)}.
\]
At first, as you know, using integration by parts we have
\begin{equation}\label{equ1}
\|u\|_{H^1(\Omega)} \leq C \|f\|_{L^2} \leq C \|f\|_{L^p(\Omega)}.
\end{equation}
Then consider a cutoff function $\eta \in C^\infty_0(\Omega)$, denote by $v = \eta u$, then $v$ satisfies the equation
\[
-\triangle v = \eta f - 2\nabla u \cdot \nabla \eta - \triangle \eta u, \quad x \in R^2.
\]
It's known that $\xi_i\xi_j/|\xi|^2$ is an $L^{R^2}$ multiplier, that is,
\[
\|\partial_i\partial_j u\|_{L^q(R^2)}  \leq C \|\triangle u\|_{L^q(R^2)}, \quad q \in (1, \infty).
\]
Using this fact and \eqref{equ1}, notice the support of $\eta$ we obtain
\[
\|u\|_{H^2{(\Omega)}} \leq C \|f\|_{L^p}(\Omega).
\]
Then the Sobolev embedding theorem yields that
\[
\|u\|_{W^{1,q}{(\Omega)}} \leq C \|f\|_{L^p(\Omega)}, \quad 2 \leq q < \infty.
\]
Proceed the above argument again, we find
\[
\|D^2u\|_{L^p(\Omega)} \leq C \|f\|_{L^p(\Omega)}
\]
as desired.