Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}_p$) that $Ext^1((\mathcal{I}_p)_p,\mathcal{O}_{X,p})\cong K$, so there exists a non trivial extension \begin{equation*}0\to\mathcal{O}_X\to\mathcal{E}\to\mathcal{I}_p\to 0 \end{equation*}where $\mathcal{E}$ is a coherent sheaf over $X$. Now I would like to prove that $\mathcal{Ext}^i(\mathcal{E},\mathcal{O}_X)\cong 0$ for all $i>0$. The case $i=1$ is clear since if $\mathcal{Ext}^1(\mathcal{E},\mathcal{O}_X)$ were not be zero, then the connecting homomorphism from $\mathcal{Hom}(\mathcal{O}_X,\mathcal{O}_X)$ to $\mathcal{Ext}^1(\mathcal{I}_p, \mathcal{O}_X)$ would be surjective and this is not possible. So my first question

How can I prove that $\mathcal{Ext}^i(\mathcal{E},\mathcal{O}_X)\cong 0$ for all $i\ge 2$?

My first attempt was to take $\mathcal{Ext}^i(-,\mathcal{O}_X)$ to the exact sequence above, but then I've realized that I should compute $\mathcal{Ext}^i(\mathcal{I}_p,\mathcal{O}_X)$ first. Here comes my second question

How can I compute $\mathcal{Ext}^i(\mathcal{I}_p,\mathcal{O}_X)$ for all $i\ge 0$?