Is it permissible for a man to be the guardian (wali) of his daughter, who was born out of an illegitimate relationship with a woman who later embraced Islam, especially given that a Muslim young man wishes to marry her and is aware of the circumstances of her birth?

The man must sincerely repent for having an emotional relationship with the woman. The girl is not attributed to him but to her mother, and he is not her guardian for marriage. The Muslim ruler or his representative should marry her off, based on the Hadith: "The ruler is the guardian of one who has no guardian."