She was honest enough to admit her mistake(s). This describes her character. It was honest of her to admit her mistakes. This describes the speaker's opinion of her action. It may be a grudging opinion. She honestly admitted that she had made a mistake. This describes what she did in one incident. "Honestly" adds little meaning other than she was truthful.
Each is correct. She made a mistake (or some mistakes) and has admitted it/them. You are giving credit for the honesty of her admission. Some people don’t admit their mistakes. My preference would be the third version.