Taken into Lothlórien, the company meets Galadriel. They share with her the sad news about the death of Balin, the awakening of the Balrog, and the fall of Gandalf.
Reliving the darkness that has overtaken his people, Gimli grows sad. Seeing this, Galadriel speaks tenderly to him, using the language of the dwarves to speak of the former glories of Moria. Hearing this praise, and in his own language, becomes a conversion experience for Gimli. As Tolkien writes:
And the Dwarf, hearing the names given in his own ancient tongue, looked up and met her eyes; and it seemed to him that he looked suddenly into the heart of an enemy and saw there love and understanding. Wonder came into his face, and then he smiled in answer.
From this point on, Gimli will declare Galadriel to be the fairest lady in Middle Earth, and will tolerate no ill word spoken against her. He becomes completely devoted to the one he once swore was his enemy.
As Fleming Rutlege points out, love and understanding, especially from one you considered an enemy, is shown here to be a profound and powerful tool of peace-making, healing, and reconciliation.
And what's more, we all have this power.