How can the questioner marry a married woman after they have committed adultery together, knowing that she is seeking a divorce from her husband due to his ill-treatment and not in order to marry him?

What you did with that woman is a heinous and grave crime, and rectifying it is not achieved by marrying her, for that would be corrupting a woman against her husband, which is one of the major sins. Some scholars have even ruled that a marriage to one who has corrupted a woman against her husband is invalid. Rectifying what you did is achieved through sincere repentance. As for the woman, no matter how bad her husband's treatment was, it does not justify her betrayal and falling into indecency. She must repent, and if she cannot remain with him honorably, she has the right to seek divorce or khul', and then you may marry her after her 'iddah (definition) has expired.