well, we are halfway through Batman/Catwoman and personally, I am starting to dought Catwoman killed The Joker.
Given batman knew he was dying I would say he tracked down joker and killed him to protect his daughter from whatever the joker had planned for when bruce died. with Catwoman taking the fall to protect her daughter from the truth. Catwoman was protecting Batman to prevent him from killing the joker.
I am enjoying the ups and downs of batcat in the early years.