I have answered this 知り合い but in answer's sheet it was 付き合い and I don't get the difference.
Here, 知り合い is not possible because of とは and があります.
彼女は昔の知り合いです would have been fine. However, here, the と after 彼女 indicates that we have to choose 付き合い, that is because the verb 付き合う takes the particle と.
As for the difference between 付き合い and 知り合い, 知り合い is a mere acquaintance whereas 付き合い implies a much closer relationship.
知り合い refers to people. It means "acquaintance" in the sense of "people who know each other". It's like "friends" but not so close as 友達.
付き合い refers to relationship. It means "acquaintance" in the sense of "the state of knowing each other". It's close to 交際.
Literally, "With her, since a long time ago, there has been 付き合い."
→ I've had 付き合い with her for a long time.
To use 知り合い:
Literally, "With her, since a long time ago, (I've) been 知り合い."
→ She and I have been 知り合い for a long time.
(i.e. She and I have known each other for a long time.)
(彼女とは昔からの付き合いです would be fine, too.)