Reflections !

Let $z$ be a length of a side of square $ABCD$ .

$m_{EB } = \dfrac{1}{2} \implies m_{AA'} = m_{\perp} = -2 \implies$ $2y - x = z$ and $y + 2x = z$

$\implies x = \dfrac{z}{5}, y = \dfrac{3z}{5} \implies \dfrac{a}{2} = \dfrac{z}{5}$ and $\dfrac{z + b}{2} = \dfrac{3z}{5} \implies a = \dfrac{2z}{5}$ and $b = \dfrac{z}{5}$

$\implies DP = \dfrac{2z}{5}, A'P = \dfrac{z}{5}$ and $ED = \dfrac{z}{2}$

$\implies A_{DEA'P} = \dfrac{1}{2}(\dfrac{z}{2} + \dfrac{z}{5})(\dfrac{2z}{5}) = \dfrac{7z^2}{50} = 14 \implies z = \boxed{10}$ .