The first woman's comment, "I am no more sick than you are," needs to be interpreted correctly. Like "got more sense," the phrase is idiomatic. Another way of phrasing the sentence "I am no more sick than you are" could be:
Neither of us is sick. I'm not sick, and you're not sick. If you think you are sick, then you're wrong. I'm healthy; you're healthy.
Then again, what the first woman is saying means something completely different if what she said were to be worded as follows:
I am not sicker than you are.
When the second lady responds, "Oh yes you are," the first lady emphasizes that her assessment is better than the second woman's assessment because her assessment makes more sense, since she has better sense than the second lady has.