Adultery does not have kaffara. The person must seek for forgiveness. It is a great sin. If it is required to be proved that the act did indeed take place, there is a need for 4 adil (just) male to give witness that they saw the act taking place (with their own eyes) and then only can the punishment be given.
Anyways there is no Kaffara and only istighfar meaning that the person is remorseful and has no intention to do it again.
Also as a side note: once someone does such an act such as adultery, they should not disclose it to anyone but just seek forgiveness. Other acts where compensation is possible such as theft, namaz, fasting, murder, it is allowed to disclose the act and find a solution such as qadha or returning the item stolen etc. However in cases such as adultery, disclosing one's sin is haraam and is a sin in itself.