"She knows me."
Translation:Ona mnie zna.
While I cant't tell you why, I can tell you how I as a native speaker "feel them" "Ona mnie zna" feels like a normal sentence , it could be stressed on any part and be said as a statement or question "Ona zna mnie" has stress on "mnie" and also feels more like a question or exclamation than normal statement.
Only the first one sounds natural. You can use the latter in some context to emphasize the object (mnie):
"Ona zna mnie, ale ciebie nie" = "She knows me, but not you".