And Sansa might be right. She has one true knight searching for her, on the old story quest to rescue the fair maid and return her to her family. Brienne of Tarth is the only living character who values honor above all else, who is determined to keep all her vows, who respects life and wants to protect the weak.
She is also, of course, an unattractive woman, despised and mocked by almost everyone she encounters. She is not technically a knight. She has all the inner qualities of the storybook hero and none of the external qualities, in a world where appearances and superficialities are all that seem to count. She is a woman who does not seem to fit anywhere in her world.
Brienne is despised for being both too much and not enough of a woman. She is not delicate enough, not beautiful enough, not passive enough for men to accept or respect her femininity, but she is still inescapably female, and so can never be anything more than a joke of a warrior to most of the men she meets. Her physical strength and fighting prowess -- traditionally male traits -- are a joke to them. Her sexuality, her (lack of) attractiveness, her naivete -- traditionally female traits -- are a joke.
Perhaps it is partly Brienne's femininity, and her inability to quite fit wherever she turns, that makes her the series' most honorable living character. Brienne suffers from prejudice and mockery, and she does not benefit from the same privileges or expectations that fill every inch of most knight's lives. She has had to fight for her position, against the hatred and disapproval of many others, and so, to succeed, she must believe, more deeply than anyone, that the efforts are worthwhile. She believes in the importance of chivalry and honor, in not killing needlessly, in protecting others, because these ideals drove her to become a knight (or something like one), and because she knows what it is like to be unprivileged, to be immediately judged as unworthy, and to be considered as nothing more than a pawn in the games of others.
But this honor is, in a way, naivete. She might be Arya-like in her desire to fight, but she is Sansa-like in her understanding of the world. Despite being a warrior, despite being unattractive, she is deeply romantic, in both senses of the word. She is innocent and sweet, if also stubborn and bold. She might be an ugly female protagonist (such a rare creature already), but she still has a heart, and she falls for a princely man who is an impossible prospect in more ways that one. Like Sansa, she believes in the stories, in the importance of honor and chivalry and bravery, and again, like Sansa, this belief sometimes feels like the result of emotional desperation, a need to protect oneself with the promises of fiction. Sansa must hope that a true knight will come for her, because she cannot escape King's Landing on her own, and Brienne must believe in the honor and value of knighthood, because she does not have a place without it. She is broken by the feeling that she is neither a son nor a daughter for her father, that she does not really belong anywhere, and so she believes, with all her heart, in the honor of knighthood, in her ability to carve a place for herself through the knightly behavior that everyone else dismisses. If she follows the code, she will be worthwhile.
But there is no space for a true knight, not even one as unconventional as Brienne, in Westeros. And so she comes up against Jaime Lannister, the handsome, golden haired, talented, oath-breaking, ruthless, immoral alternative to her unattractive, unconventional honor. He is Brienne's inverse: the true, storybook knight on the outside, corrupt and selfish on the inside... yet his version of knighthood shows an understanding of Westeros that Brienne lacks. Vows and honor conflict. Morality is not black and white. And so, just as Brienne reminds Jaime about the importance of honor and loyalty, his presence in her life forces her to face the fact that storybook knighthood is unfeasible in harsh reality.
Yet if there are no true knights in Westeros, there is space for a different kind of hero. Brienne's story is an adaptation of a traditionally male narrative, one that usually sidelines or victimises female characters. She swears fealty to a woman, as male knights swear to their liege lord, because she respects that woman's strength, her bravery and her kindness. She goes on a quest to save the beautiful maiden, but not to marry her or benefit from the quest in any way, but to return her to her mother. Because she cares for Catelyn, and because it is the right thing to do. It is a story of a woman, rescuing a woman, for the sake of another woman. It is a rare story where the mother, the young girl and the shieldmaiden are all given equal weight and worth. Brienne, despite taking on many stereotypically male traits, is not "one of the boys" or in any way dismissive of her gender as a group. She does not fit into the role that society has assigned for her, but she does not disparage those who do. She uses her strength and her skill to respect and help other women in ways that most men in Westeros would never even think to attempt, because she understands, more than any other knight, that women are truly worth something as individuals.