In triangle $ABC,$ $M$ is the midpoint of \(\overline{AB}\). Let $D$ be the point on \(\overline{BC}\) such that \(\overline{AD}\) bisects \(\angle BAC\), and let the perpendicular bisector of \(\overline{AB}\) intersect \(\overline{AD}\) at $E.$ If $AB = 44$ and $ME = 12,$ then find the distance from $E$ to line $AC.$
