Uh, it's a bit complicated, but "sexist" is a bit too harsh. It's not sexist, the way I understood it, you just "corrected" her because she used the term used for males for a female. I'd call that sexist, rather than the opposite. The word "landlord" is obviously from a time when women couldn't own lands, but now they can, so why not use the term "landlady"? Honestly, in my language, it would have been incorrect to accompany the male noun with a female name (our nouns have gender; male, female and... middle gender. It's complicated. Also, our "my" has both male and female versions (and middle gender version) so it too would have to be in appropriate gender). The owners of my apartment that I rent when I'm in college are a man and a woman; when I refer to the man, I call him the "landlord" and when I refer to his wife, I call her "landlady". It would be weird and senseless to say it otherwise, at least in my language. Just recently I had to explain to one of my friends that my landlady will wait for them in front of the building; if I used the word "landlord", my friend would expect to see a man and would not recognise the person I was referring to and it would be confusing. I had to use the noun reserved for females to avoid confusion and for my friend to know whom to approach. In my opinion, it's perfectly normal to make a distinction, at least grammatical one, like that (if such distinction exists; for example, the word "boss" is used for both genders and doesn't have a distinction. Since both "landlord" and "landlady" are legit words, they are okay to use when you're describing a person).
Honestly, I would've been a bit surprised too if I heard the phrase "My landlord Chloe". What now, "My husband Chloe"? "My brother Chloe"? When there's a specific word for both male and female, we use the appropriate word for appropriate gender. That's not sexism, that's grammar. Sexism would be if you laughed at the fact that a woman "owns a land" and demanded to see a man to talk with about the issue. How is hiding behind old, patriarchal terminology and refusing to admit that a woman is a woman better than acknowledging and accepting that a woman holds a, formerly, exclusively male position? Some people's logic.