What is the ruling on my continuing with my second wife after she committed adultery and repented, and what is my sin and its expiation in this situation?

The husband must be just among his wives in providing for them, in lodging, and in every apparent matter. Whoever is unable to be just is not permitted polygamy, unless one of them waives her right. If the second wife repents, it is permissible to keep her on condition of justice or her waiving her right. If her 'iddah (definition) (waiting period) has expired, she is not lawful except with a new contract. The expiation for lack of justice is repentance to Allah, seeking forgiveness from the wronged wife, and pleasing her, along with improving one's conduct in the future.