Is the failure of the woman who was fornicated with to repent considered a sin upon the questioner? And what must he do after his deed, knowing that he feels deep remorse, is distracted in his prayers, has forgotten what he memorized of the Quran, and is unable to give lessons?

You have committed adultery, which is a heinous sin and one of the major sins. However, Allah accepts the repentance of those who repent and rejoices in it, so do not despair of His mercy. It appears from your question that you are remorseful, and remorse is repentance. Increase your acts of seeking forgiveness (Istighfar) and performing good deeds. The woman's failure to repent does not harm you, for every soul is responsible for its own actions. Do not resign from your job, and strive in what benefits you. Occupy yourself with learning beneficial knowledge and reviewing the Holy Quran. Do not refrain from leading the prayers (Imamah) if you are qualified, for one who repents from a sin is like one who has no sin. Turn away from whispers and doubts, and seek refuge with Allah from Satan.