Has there ever been an exorcism conducted by the Mormon Church? Has there ever been a recorded exorcism conducted under the authority of the Mormon Church?
Exorcisms aren't recorded in the LDS Church, in fact the ordinance itself isn't even documented, it's known unofficially as one of the, "Unwritten Ordinances" of the church, along with dusting your feet (see Matt 10:14) and other similar ordinances.

There are many stories which exist as Mormon urban legends, mostly involving missionaries serving in highly superstitious societies. I heard many serving as a missionary, one particularly prominent tale is of a, "haunted house" somewhere in South America, the juste of the story is there were many evil spirits dwelling in a single residence, and when the Elders commanded them to leave there was audible screaming and all the windows and doors in the house opened and slammed shut.

I myself was called to the homes of a couple families of ethnic origin (Haitian and Puerto Rican) because of paranormal activity. One was on account of a sighting of an evil spirit flying around in a kitchen, sticking it's tongue out at people, the other was to the home of a woman who was acting... not like herself/possessed. Both were very unsettling experiences, but neither resulted in casting out devils, though the thought had crossed my mind, instead we offered a blessing on the homes, which seemed to suffice in both instances.

The only officially documented instance of an exorcism is found in, History of the Church Volume 1:82-83, and is recorded because it was the first miracle performed in the restored church (read The Knight Family: Ever Faithful to the Prophet). The account goes like this:


  April 1830 In the month of April, 1830, Joseph Smith was visiting at
  the house of a Mr. Joseph Knight, at Colesville, Broome County, New
  York. This gentleman had rendered the prophet some timely assistance
  while translating the Book of Mormon, and he was anxious that Mr.
  Knight and his family should receive the truth. While in Mr. Knight's
  neighborhood the prophet held a number of meetings. Among those who
  attended regularly was Newel Knight, son of Joseph Knight. He and the
  prophet had many serious conversations on the subject of man's
  salvation. In the meetings held the people prayed much, and in one of
  the aforesaid conversations with the prophet, Newel Knight promised
  that he would pray publicly. When the time came, however, his heart
  failed him, and he refused, saying that he would wait until he got
  into the woods by himself. The next morning when he attempted to pray
  in the woods, he was over-whelmed with a sense of having neglected his
  duty the evening before, in not praying in the presence of others. He
  began to feel uneasy and continued to grow worse both in body and
  mind, until upon reaching home his appearance was such as to alarm his
  wife. He sent for the prophet, who, when he came found Newel in a sad
  condition and suffering greatly. His visage and limbs were distorted
  and twisted in every shape imaginable. At last he was caught up off
  the floor and tossed about most fearfully. The neighbors hearing of
  his condition came running in. After he had suffered for a time the
  prophet succeeded in getting him by the hand, when Newel immediately
  spoke to him, saying he knew he was possessed of the devil, and that
  the prophet had power to cast him out. "If you know I can, it shall
  be, done," replied the prophet; and then almost unconsciously he
  rebuked Satan and commanded him to depart from the man. Immediately
  Newel's contortions stopped, and he spoke out and said he saw the
  devil leave him and vanish from sight. (Joseph Smith Healings/Miracles)


The rest are kept secret, as are most miracles performed under the authority of the Priesthood.


  36 And he charged them that they should tell no man... (Mark 7:36)