I have a daughter from my previous marriage who never accepted Islam and she now has an illegitimate child. As a practicing Muslim how am I supposed to act towards this child?
You need to keep in touch with her and treat her nicely being a practicing Muslim so that she will see the great manners of Islam. You should try to invite her to Islam as and when you find a suitable chance. Her being non Muslim does not mean always that she denied Islam deliberately. She might had been a victim of ignorance or misleading propaganda.
You should do your best to give the practical evidence that Islam is the best religion.
Wassalam.