You have been acknowledged a friend of Lothlórien through your efforts on behalf of the Elves, and should travel to the Hill of Cerin Amroth to affirm that friendship.
'You are welcome within Lothlórien, ${PLAYER}, for the friendship you have demonstrated and your efforts against the Orcs and other dangers within Moria.
'Pass beneath the eaves of the forest, to the north-east, and travel to the hill of Cerin Amroth in the centre of Lothlórien; it is both beautiful and peaceful, and Milwen will be pleased to tell you some of its history. Look for her by the archway west of the hill of Cerin Amroth, far to the northeast of here inside the forest.'
If you have not already completed Volume II, Book 6, it is encouraged, though not required, that you do so before beginning this quest.
Milwen is by the archway west of the hill of Cerin Amroth, located in the centre of Lothlórien, far to the north-east of Talan Haldir and east of Echad Andestel.
You have been told to speak with Milwen on the hill of Cerin Amroth, within the woods of Lórien.