Is it permissible to marry a girl after a previous relationship with her mother that did not exceed the bounds of friendship, given the repentance of both parties and the questioner's knowledge of the girl's uprightness and her family's commitment?

There is no impediment to completing the marriage. Touching a non-mahram (definition) woman with desire, without intercourse, does not lead to the prohibition of marriage according to the majority of jurists (Malikis, Shafi'is, and the relied-upon opinion among Hanbalis). However, the Hanafis hold that touching a woman with desire prohibits her daughter and mother [from marriage], because it is a cause leading to intercourse. But the touching must be with a recognized desire (erection or increase in erection) and looking at the inner private parts. As for ejaculation by touching, it does not necessitate prohibition.