This mission is basically a boss battle with Lilith and her Mundus Spawn. First, you have to pull yourself towards the exposed eye on the boss's side and keep attacking it until the main eye on its head opens. Once its open, keep yourself airborne and attack the eye continuously. Just make sure to use angel boost or aerial evade to avoid getting hit by its flailing arm. Once enough damage is dealt, the spawn will keel over, exposing its tummy. Pull Lilith out and pummel her.
Later in the battle, the spawn will close its other eye so you have to use demon pull to expose it again. Repeat the same process as before, using angel lift to get close to the eyes and damage them. Every time the spawn falls down, you have to pull Lilith and continue damaging her until she's done for.
She is nothing without the brat