It was suggesed that she was extremely infatuated with Batman to the point that she assumed the costume identity in order to get close to him and hopefully engage in an emotional relationship with him. It worked but not quite in the way she had hoped, as the Batman wanted her to retire from the role for his fear that she would be hurt in her costumed exploits. Kane would continue to operate in her costumed identity until around the early-1950s when she learned that Batman had become deeply emotionally interested in someone in his unmasked identity.
While Wayne never revealed his identity to her, or explicitly told her of his private life, she said she knew of his change in his attitude to her and assumed as such. She largely retired from her costumed role soon after this, though she continued to "help out" Batman on occasion, not realizing that Grayson had replaced Wayne in the role.
She learned that Wayne was the original Batman when his costume identity was revealed to the general public with the death of Wayne's wife. It was not revealed if Wayne ever returned her feelings after the death of Selina but it was suggested that Wayne did for she lamented about her feelings for Wayne when the Earth-One Wayne came to Earth-Two suggesting that she and the Earth-Two Wayne had finally revealed their feelings to each other directly. Kane realized that he was not the love of her life reborn, but was a completely new person. She finally realized that her chance at loving Batman was over and retired from the role and settled down to a civilian life.
If she had not died in the intervening years since that meeting with the Earth-One Batman, she was killed along with everyone else on his native Earth-Two in the Crisis on Infinite Earths and completely erased from history.