What is the ruling on one who wrote a letter containing slander, without intending to cause offense, then denied it and swore by the Qur’an that she did not write it, and is now remorseful and seeking to repent?

Swearing by Allah falsely is one of the gravest prohibitions, and one must repent to Allah for it. Scholars have differed regarding the necessity of expiation for the false oath (yamin al-ghamūs), but the more cautious approach is to offer it. One must have a good opinion of Allah, and repent from slander as explained in previous fatwas.