Is a marriage contracted without witnesses and without the family's knowledge valid in Islamic law, and must I give her the Qur'an and divorce her if she insists on separation?

The aforementioned marriage is invalid due to the absence of the legal conditions such as a guardian (wali) and two witnesses. The agreed-upon dowry (mahr) must be paid, unless the woman waives it. What transpired is not a legal marriage; rather, it is fornication (sifah). One must repent to Allah and sever ties with the woman. If you wish to be legally bound to her, you must seek her hand from her guardian. Istikhara (prayer for guidance) is not permissible for a forbidden relationship.