I think she is his half-sister but they lived in different houses. Then, I think she fell in love with him when she was feeling down and felt that she was special for the Duke. The kindness and generosity of the Duke deeply affected her desire too deep so she was obsessed with the Duke.
There were cases that when they have siblings (half or biological) but live in different houses then they reunite after so many years, there are possibilities that they will attracted to each other as not siblings, but lovers (or one-sided).
There were cases that when they have siblings (half or biological) but live in different houses then they reunite after so many years, there are possibilities that they will attracted to each other as not siblings, but lovers (or one-sided).