Lothlorien may feel like being within a song, but the elves here do not exhibit the same joyful and playful musical inclinations of their brethren in Rivendell. The whole of Caras Galadhon is great and very fey. The Lady Galadriel knows more than news brought by courier and word of mouth, and Lord Celeborn could be a diplomat with the care and recognition he takes to acknowledge each of their guests.
The elves here know exactly what it means for Gandalf to have fallen, as they also know the threat of a Balrog. The danger and loss causes Celeborn to speak without thought, blaming the dwarfs for their delving and rash action on part of the Fellowship. Galadriel, in her near omniscience, calls Celeborn on his words and wins the adoration of Gimli. The wisdom of Galadriel goes beyond reading the unspoken thoughts and cautioning against speaking from fear or anger. Saruman's leadership of the White Council is contrary to her efforts, and as she reflects, things may have progressed quite differently under Gandalf's leadership.
Last chapter I commented on the lack of expressed grief, but our Fellowship and the elves mourn within the safety of the elven wood. The matter too raw for Legolas to translate the mourning songs of Lorien, and strong enough to move the rarely lyrical Frodo to his own song. I do understand that before they were under enough threat that they didn't want to stop and mourn, but that does stand a bit counter to all the time taken to explicitly admire the landscapes. Frodo's comment that "I don't miss Gandalf's fireworks, but his bushy eyebrows, and his quick temper, and his voice," strikes me as perhaps one of the most poignant reflections on loss that I've read. Yes, his magic was useful, but that's not what really matters when it comes to grief.
Galadriel sees what is unspoken and what is yet to come, through her own power and through tools within her reach. In particular, she reads and speaks of secret desires, not of lust, but those to admit to may be a sign of cowardice. For the hobbits, their secret desires are not so unknown, but simply to return to the comfort of the Shire and the lives they once knew. Boromir won't speak to his, but even without foreknowledge and familiarity with the story one can guess his secret desire speaks to his pride and wish for power. Her mirror itself shows much, of Gandalf's impending return, of conflict to come, forces moving, and of the Eye ever seeking the Ring.
The Lady also speaks of consequences and willing sacrifice, that the saving of Middle Earth means the diminishing of the elves. They are truly facing an end of their age, no matter the end of this war.
I do want to commend how well they matched Cate Blanchett to Alan Lee's illustrations, and Lothlorien is gorgeously recreated. Blanchett isn't quite as fey and ethereal as my mind image of Galadriel, but that's the issue with headcannon. I picture someone more like Tilda Swinton who always looks fey-touched to my eyes.
The elven mourning song that Legolas declines to translate rings high and pure, more something to be experienced than to pick out the words. Here Boromir does speak somewhat of Galadriel's voice in his mind, of his fear and his hope, while Galadriel speaks openly of the subtle strength the party holds.
Visually, Galadriel and Frodo at the mirror only lacks the companionship of Samwise to match the book and the artwork of Alan Lee. The visions themselves, and interpretations differ greatly however. There is no vision of one who could be Gandalf, but instead visions of the Fellowship splitting and of the Raising of the Shire. Galadriel explicitly states that the Fellowship is splintering, and that one will attempt to take the ring, unlike her role as vision guide rather than interpreter in the book. I don't like the 'dark queen' bit, and even before the special effects aged it never struck me as quite right, but Blanchett nails the accepting of her fate and encouraging frodo.