Since the torque about the vertical axis is zero, we can conserve angular momentum about the vertical axis.
m v d = I w + m d2 w -------------- (1)
Where 'm' is the mass of the bullet, 'd' is the distance of the center of the door from the axis, 'w' is the angular speed after impact, 'v' is the velocity of the bullet before impact, 'I' is the moment of inertia of the door about the axis of rotation.