While I was still a non-Muslim, I fathered an illegitimate girl.

Answer

A paternity test is not accepted as Islamic evidence even if it is 100%. Besides this, even if the man and the woman accept that it is his child and there is no doubt whatsoever in this fact, then too the Shariah decrees that he is not the legal father. Bear in mind that the Shariah does not deny biological paternity. Whilst accepting that the man is the biological father, the Shariah says that he is not the legal father because rights (huqooq) are the products of Nikah, not of zina (fornication). Thus, you will not be the legal father even if the tests should establish 100% you are the biological father.

According to the Hanafi Math-hab, you will be her mahram. But according to the Shaafi' Math-hab you will not be her mahram. Regardless of what the kuffaar determine with their technology, the Shariah in terms of the Shaafi' Math-hab does not change. All rules applicable between a ghair mahram man and a woman will apply to you and the girl. Morally, you may support her financially, but you are not her mahram if you are a Shaafi'.