How can one atone for the sin of wronging a teacher and swearing falsely, especially after an apology that was not accepted, and with the impossibility of informing one's parents or the administration?

Repentance has conditions, including: abandoning the sin, regretting it, and resolving not to return to it.

If the sin involves a human right, then it is a condition to restore the right or seek the forgiveness of its owner, based on the saying of the Prophet, peace and blessings be upon him: "Whoever has wronged his brother concerning his honor or anything else, let him seek his forgiveness today before there is no dinar and no dirham."

If the injustice against the teacher was financial, then it is obligatory to return the money unless she forgives it. If the injustice related to her honor, then it is obligatory to apologize and seek forgiveness. If that is difficult or would lead to a greater harm, then it suffices to mention her positively to those who heard her backbiting.

As for swearing a false oath, repentance from it is obligatory, and it is more cautious to offer expiation for an oath.