Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
As long as this woman has repented and has been feeling deep sorrow for her disobedience to Allah, Allah will forgive her sin since true repentance erases all the past sins. A person who makes repentance is just as the one who did not commit a sin at all. Allah has mentioned in the Qur'an that the one who associates partners with Allah, or kills a person or commits fornication and makes repentance Allah forgives them. Allah Says (interpretation of meaning): {And those who invoke not any other ilâh (god) along with Allâh, nor kill such life as Allâh has forbidden, except for just cause, nor commit illegal sexual intercourse and whoever does this shall receive the punishment. The torment will be doubled to him on the Day of Resurrection, and he will abide therein in disgrace; Except those who repent and believe (in Islâmic Monotheism), and do righteous deeds, for those, Allâh will change their sins into good deeds, and Allâh is Oft-Forgiving, Most Merciful.}[25:68-70]. Allah Loves those who return towards Him in repentance. Allah Says (interpretation of meaning): {Truly, Allâh loves those who turn unto Him in repentance and loves those who purify themselves (by taking a bath and cleaning and washing thoroughly their private parts, bodies, for their prayers, etc.).}[2:222].
Hence, she should cover herself and should not inform anyone about her sin. Ubadah Ibn Al-Samit reported that: 'Allah's Messenger (peace be upon him) said: "Receive (teaching) from me, receive (teaching) from me. Allah has ordained a way for those (women). When an unmarried male commits fornication with an unmarried female (they should receive) one hundred lashes and banishment for one year. And in case of married male committing adultery with a married female, they shall receive one hundred lashes and be stoned to death." [Muslim]
There is a Hadith that reads: "Whoever commits a sin in this life and gets its punishment in this life, Allah is the Most-Just, so, He will not punish him again (in Hereafter). If a person commits a sin in this life and Allah covers him in this life then Allah is Most Generous so he will not punish him in the next life." [Ahmad, al-Arnawoot consider it as Hasan]
In the light of the above details you may have understood the verse of Surah al-Furqan and the position of this woman who had committed illegal sex.
The verse 15 from Surah al-Nisa is abrogated, the evidence is the aforementioned Hadith of Ubadah Ibn al-Samit.
As for the verse 16 from Surah al-Nahl, it was also abrogated as reported by Imam Al-Qurtubi.
Some Hudood (legal punishment) are established by Qur'an as in the case of giving lashes to a fornicator which is mentioned in the second verse of Surah al-Noor while some other Hudood are mentioned in Ahadith as in the above Hadith of Ubadah.
As for the illicit child he is traced back to his mother contrary to the father. For more details in this regard, please read the Fatawa: 84788.
Know that if a married woman commits adultery whether before pregnancy or after it, the child is traced back to her husband as the Prophet (Sallallahu Alaihi wa Sallam) said: "The boy is for the owner of the bed and the stone is for the person who commits an illegal sexual intercourse." [Al-Bukhari]
A husband may only deny his affiliation to his child by taking an oath in front of authorities which is called Lia’an.
We ask Allah to accept her repentance and grant her comfort, especially if she was not aware of this sin.
Allah knows best.