"but I can't spend my life like this"

If you want to be in HIS life, then you have at least 9 more years of this until she is of age, and then some. He had a child with this woman. She is the child's mother. She will ALWAYS be the mother.

I've no children of my own, but my wife has two. It's a difficult situation to deal with, but if you chose to stay with him, there is no choice. Accept it, and find the least dramatic way to deal with it, for the sake of the child at least, if not for your own sanity, too.