How should a child born out of wedlock, who is the sister of a Muslim wife and whose father is not Muslim, be dealt with?

The child born of adultery is not attributed to the adulterer, but rather to its mother and inherits from her. Therefore, the daughter born of adultery is not a relative of your wife, nor is she a forbidden relative (mahram) to your son, and she is treated like any foreign woman upon reaching maturity.