What is incumbent upon me regarding my illegitimate daughter and her mother, especially since my wife knows and is demanding divorce if I marry the child's mother?

You must repent from adultery. The illegitimate daughter is not attributed to you, and you are not obligated to spend on her or her mother. However, spending on them is an act of kindness for which you will be rewarded. Your wife does not have the right to prevent you from spending on them. Lying is permissible to preserve marital harmony, provided that no rights are violated; however, dissimulation (Tawriya) is preferable. There is no harm in marrying an adulterous woman if she repents and becomes righteous, and if you are capable of being just. Your wife does not have the right to demand a divorce merely because you wish to marry another, and polygyny may not always be in one's best interest.