What crime is Alice Morgan charged with?

Alice is incarcerated in the mental hospital because she confessed to the murder of John Luther's corrupt colleague and false friend DCI Ian Reed. We can infer that Alice was charged with his murder, although there were likely extenuating mental circumstances. A secondary explanation for her time in the mental hospital is her suicide attempt. There is no suggestion that she has confessed to the murder of her parents.

At the conclusion of series/season 1, John Luther catches up with Reed, who had murdered Zoë, Luther's ex-wife, and framed Luther for the murder. Luther extracts a confession from Reed at an empty train station. Preferring death to imprisonment and disgrace, Reed taunts Luther, claiming that he and Zoë had had an affair, and tries to enrage Luther to the point of killing him. However, Luther refrains, but, in a struggle, Reed stabs Luther. Alice, who, along with Zoë's grieving boyfriend Mark, had assisted Luther and met him at the station, steps in with a gun and says that she would like to kill Ian as punishment for his crimes. Wounded, Luther objects but is unable to stop her. Alice, looking for a tie-breaker, asks Mark for his opinion. Mark agrees with Alice that Reed should die. Alice shoots Reed just before the police arrive.

Although the show does not cover the immediately following scenes, series/season 2 opens with Alice being interrogated by DCU Schenk in the room that is later identified as an interview room in the mental hospital. Alice flatly states, referring to DI Reed, "He was a friend who betrayed John and hurt him very badly, so I killed him. Two barrels of his own shotgun. Boom, boom." She explains that Luther did not ask her to kill Reed; in fact, Luther was angered by her decision. Alice assumed all guilt for the killing of DCI Reed.

Later in the same episode (series 2, episode 1), when Luther visits Alice in the mental hospital, her wrist is bandaged. She alludes to a suicide attempt and makes comments suggesting that the failed attempt led to her institutionalization, so that she would avoid further harm to herself. Unfortunately, the series never fully shows Alice's plot arc. We are left to wonder whether she was put in prison for the murder of DCI Reed and made her suicide attempt there before being transferred to the hospital or she was institutionalized shortly after the events of series 1.