I do feel however that Bani is not nishikant's wife's daughter, she is however nishikant's illegitimate child. (He must have been forced to marry, and later found out that the girlfriend is pregnant with Bani and since he loved her so much he took Bani in to give her his name.) I think he must have gone for a meeting with his x girlfriend. And while their occassional meetings they happened to be at Park Road where this accident took place. So the cause of his first love's death he took to being an alcoholic and gambler. As far as the checks are concerned, they were Roshni's way of compensating for the accident. And since Nishikant was not a greedy man he never wanted the 'blood money', so he donated it to the hospital. And for Jai he considered bribing for a murder something against his conscience and that's why he's so uncomfortable about it, since he can't admit legally to the accident and yet feels guilty for having something to do with the accident.
And if you notice he never cared about Pia knowing the truth, he's only concerned with Bani not finding out the truth. That shows that Bani is infact the daughter of the woman who got killed at Park Road.