First of all, yes, Miss Sadie's leg infection is totally gross. But, nasty as it is, it's also symbolic. See, she cuts herself getting Abilene's compass, which is really her compass, which she gave to her son, Ned. And that's exactly the point: she gets a physical wound right alongside the emotional would of all those painful memories.
And just like the leg wound symbolizes her emotional wounds, the healing of the leg wound symbolized—you guessed it—the healing of her emotional wounds. So when Miss Sadie tells her the first part of her story, Abilene gets some ointment for her leg. By listening to the story of Miss Sadie's past, Abilene has started helping to heal the wound.
But not so fast.
The infection gets worse and worse as the story progresses (we'll spare you the gory details). While telling the story helps some ("It was as if the story was the only balm that provided any comfort" [18.29]), it doesn't cure it ("There is too much sickness inside and it festers" [35.6]). In order to heal completely, Miss Sadie will need to let everything out and finish the story in its entirety.
So that's what she does.
And only then does she let Abilene treat her: "Kneeling beside her, I held the hot blade to her wound and pierced it, letting all the pain flow out." (40.8) Finally, her troubled heart—and her leg—can heal.