Is it permissible to marry a woman with whom one had an illicit relationship, and she has repented from it, knowing that part of her past has been exposed to close friends?

You both did well by repenting from the forbidden seclusion (khalwa). You must cut off communication with the girl. The girl erred by revealing her sin, for it was her duty to conceal her own faults. You erred by informing your friends about her past; if that involved accusing her of immorality (fahisha), then the sin is even greater. You must repent from all of this and supplicate for her well-being. Marriage between those who love each other is religiously desirable, and the girl's past does not prevent marriage to her if she has repented and become righteous. If you fear embarrassment from marrying her due to your friends knowing her past, then you are permitted to leave her, and breaking off an engagement is permissible when there is a need for it.