How can one atone for the sin resulting from mingling and dancing with one's fiancé and other guests, and having that recorded on a video widely distributed? Is repentance and seeking forgiveness sufficient, or must the video be retrieved?

A great many transgressions and several forbidden acts occurred: appearing before a non-mahram (definition) fiancé in the manner you described, dancing, intermingling with non-mahram men, filming the celebration, and giving the tape to the fiancé's family, which would lead to men seeing non-mahram women. Since you have repented with sincere repentance, rejoice in Allah's acceptance of your repentance and His forgiveness, for repentance erases what came before it. As for the tape, if you can retrieve it, then do so. Otherwise, there is no blame upon you, for Allah the Almighty says: "So fear Allah as much as you are able."