How can one atone for the sin of communicating with a married woman online and the ensuing lies told to mend relations, after her husband discovered the matter and confronted me?

Islam has set controls for dealing with foreign women, and the temptation of women is among the greatest temptations. Simply speaking with a foreign woman is a door to corruption, and your communication with her is a reprehensible act. It is not justified by being for the sake of brotherhood in Allah; rather, it could be one of Satan's tricks. Lying is forbidden, but it may sometimes be permissible for an overriding benefit, such as reconciliation and preserving family unity.