Emma is used to people who are sick, or prone to sickness, like her father and sister. And when Jane Fairfax, whom she has always been prone to dislike, shows up a little thin, with the excuse of ill health, Emma at first resolves to dislike her no more.
Then the old grievances settle in, and we only see bits of relenting in Emma's attitude, usually accompanied by Jane Fairfax's discomfort (at the Coles's about the piano forte, or when Frank Churchill improperly teases her at the Bates's the next day), until Jane's sick headaches upon accepting the governess position become violent.
Then, Emma becomes compassionate, kind, overt in her advances of friendship. Does anyone think that Emma relented, not only because Jane Fairfax was entering a life of privation, and Emma had had a recent wake-up call about her own unfeeling attitude, but because Jane Fairfax was a suffering patient, and Emma knew how to relate to her on that level?