AMC 10 2021b Problem 20

The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$

$\textbf{(A)} ~20 \textbf{(B)} ~21 \textbf{(C)} ~22 \textbf{(D)} ~23 \textbf{(E)} ~24$