jmc

Since $5^2+12^2=13^2$, triangle $AEB$ is a right triangle. Define $F$ to be the foot of the perpendicular drawn from $E$ to side $AB$. The distance from $E$ to side $AD$ is $AF$. By the similarity of triangles $AEF$ and $ABE$, $$\frac{AF}{AE}=\frac{AE}{AB}.$$Solving for $AF$, we find $AF=A{E}^2/AB=5^2/13=\boxed{\frac{25}{13}}=\boxed{1\frac{12}{13}}$ units.