Question

Then he turned his back on me, and went on talking to his wife; and I knew what that meant, too. I was not the kind of morsel he was after. The typhoid had served me well enough in one way: it kept that kind of gentleman at arm’s-length. What does the narrator mean when she refers to Mr. Brympton as “that kind of gentleman”?

Answer 1

She refers that : She does not trust Mr. Brympton.
That's why the narrator made an effort to keep Mr. Brympton away so he's not to close to her

Answer 2

The narrator means that she does not trust Mr. Brympton.

The narrator describes a situation in which Mr. Brympton turned his back on her, which is a sign of rejection, and later mentions the good part of having "typhoid", which it's a disease characterized for affecting the physical appearance (Eruption of red spots in the skin). The disease keeps that "kind of gentleman" away. When she says that "kind of gentleman", she refers to the kind of men that are superficial and prejudiced, whose behavior changes based on how people look, and she doesn't trust that "kind of gentleman".