Will a woman who falsely slandered another woman and insists on stubbornness and denial enter Hell, knowing that the slandered woman graciously relinquished her right? Is this relinquishment sufficient, or must the slanderer's family give charity or expiation on her behalf?

The transgressor against honor must repent to Allah and seek forgiveness from the one she wronged. Her repentance is not complete unless she seeks forgiveness from her victim, if she is able to do so. Forgiving her means that Allah will not punish her concerning your right, but Allah's right remains, and His matter is for Him. It is good for her guardians to seek forgiveness for her and give charity on her behalf.