Everything that Stevens isn't, Miss Kenton is. He acts robotic and she acts like a flesh-and-blood human. He's practically emotionless, and she has tons o' emotion. He has repressed his romantic and sexual urges, and she understands that she had needs, dagnabbit.
Sigmund Freud once famously suggested that we are all divided into an ego (the conscious self that tells you to get to work each morning) and the id (the unconscious that just wants to sleep, have sex, eat ice cream, and beat people up). If Stevens is all stiff ego, Miss Kenton is the id. She's expressive, emotional, out-spoken… but still incredibly good at her job. Miss Kenton voices the emotions that Stevens keeps such an uber-tight lid on. And she's annoyed with him for being so bottled up:
"[…] Why, Mr. Stevens, why, why, why do you always have to pretend?" (6.50)
Miss Kenton eventually gets fed up with Stevens's stiff upper lip and leaves Darlington Hall.
She is super torn between the professional life she found so fulfilling (with the side benefit/curse of her crush on Stevens) and the personal intimacy she longs for.
This conflict is the most obvious on the evening she makes her momentous decision to marry one of her former co-workers, just as Stevens is helping Lord Darlington host an important diplomatic meeting. She doesn't seem happy about the prospect of marriage, and she can't stand the fact that Stevens appears indifferent to her big news.
Miss Kenton also speaks out about the moral dilemmas Stevens is unable to confront head-on. When Stevens asks her to fire her Jewish maids, Miss Kenton explicitly denounces the incident:
"I am telling you, Mr. Stevens, if you dismiss my girls tomorrow, it will be wrong, a sin as any sin ever was one, and I will not continue to work in such a house." (6.26)
But Miss Kenton doesn't have a backbone of steel, and she sticks around. However, she's not afraid to fess up to being all bark and no bite.
"It was cowardice, Mr. Stevens […]. Whenever I thought of leaving, I just saw myself going out there and finding nobody who knew or cared about me." (6.43)
Ooh, wow. She also confesses to a little more than just cowardice—she confesses that she thinks Mr. Stevens cares about her. But Stevens doesn't pick up on this. When Stevens also admits that he felt the decision was wrong, Miss Kenton takes him to task for not sharing his feelings with her earlier:
"Do you realize, Mr. Stevens, how much it would have meant to me if you had thought to share your feelings last year? […] I suffered all the more because I believed I was all alone." (6.50)
There's a pattern here. Miss Kenton's devotion to Stevens seeps through in many of her comments to him: she believes he cares about her, him "sharing his feelings" would have meant a lot to her, and the idea of unshared feelings makes her feel "all alone." Miss Kenton, a.k.a. Id Woman, spews out almost more Freudian slips than we can keep up with.
As we delve into their relationship, we can't help wondering whether Stevens and Miss Kenton are in fact "in love" with each other. They never come out and say so explicitly. In the moment when they are physically closest—Miss Kenton has (gasp!) snatched a book out of Stevens's hand!—Stevens tells us that they moved to "another plane of being" (6.120).
There are so many layers of emotional repression and professional distance in Stevens that we can never really know his true feelings. That's one of the frustrating, and fascinating, things about getting the story only from his perspective.
Stevens comes closest to confessing his feelings when he explains that his heart was broken that Miss Kenton would not be returning to Darlington Hall. But the expression is just generic enough that he could be talking about a good friend rather than a romantic interest. When Stevens finally meets Miss Kenton after so many years, we can't help but feel his own sadness as he notes that her "spark" had gone (8.9). But again, is this the "spark" of romantic attraction or professional excellence?
Miss Kenton explains that her marriage was troubled at first because she missed her professional life, but she has grown to love her husband and doesn't intend to leave him. Stevens and Miss Kenton leave each other good friends, but although she seems to have accepted her own advice—that there's no use looking back on the past—we have to wonder if Stevens's heart is breaking over her marriage or over her transformation.
Our money's on a little of both.