High Anxiety

The story begins at Los Angeles International Airport, where Dr. Richard Thorndyke (Mel Brooks) has several odd encounters (such as a homosexual man disguised as a police officer). He is taken by his driver, Brophy (Ron Carey) to the Psycho-Neurotic Institute for the Very Very Nervous, where he has been hired as a replacement for Dr. Ashley, who died mysteriously, possibly by murder. Brophy has a condition of nervousness, and he takes pictures when he gets nervous. He also has a hard time picking up objects that are heavy he can't life them past his knees, even though he think he can. Upon his arrival, he is greeted by the staff, Dr. Charles Montague (Harvey Korman), Dr. Philip Wentworth (Dick Van Patten) and Nurse Charlotte Diesel (Cloris Leachman). When he goes to his room, a large rock is thrown through the window, with a message of welcome from the violent ward.
Thorndyke hears strange noises coming from Nurse Diesel's room and he and Brophy go to investigate. Diesel claims it was the TV, but it was actually a passionate session of BDSM with Dr. Montague. The next morning, Thorndyke is alerted by a light shining through his window. It is coming from the violent ward. Dr. Montague takes Thorndyke to the light's source, the room of patient Arthur Brisbane (Charlie Callas), who, after suffering a nervous breakdown, thinks he is a Cocker Spaniel.
Wentworth and Diesel argue about whether he can leave the institute. After she lets him go, he drives home, but the car has been rigged to blast rock music loudly through the radio. Wentworth is trapped in his car, his ears hemorrhage, and he dies from a stroke, aggravated by the loud music.
Thorndyke and Brophy travel to San Francisco, where Thorndyke is to speak at a psychiatric convention. He checks in to the atriumed, vertigo-inducing Hyatt Regency San Francisco, where much to his dismay he is assigned a room on the top floor, due to a reservation change by a "Mr. MacGuffin". He pesters the bellboy (Barry Levinson) with repeated requests for a newspaper, wanting to look in the obituaries for information concerning Dr. Wentworth's demise. He then takes a shower, during which the bellboy comes and in a frenzy mimics stabbing Thorndyke with the paper while screaming "Here's your paper! Happy now?! Happy?" The paper's ink runs down the drain.
After his shower, a woman bursts through the door; she is Victoria Brisbane (Madeline Kahn), the daughter of Arthur Brisbane. She wants help regarding her father. Thorndyke agrees to the terms, but then finds out Nurse Diesel's plot. The patient is not the real Arthur Brisbane.
To stop Thorndyke, Diesel and Montague hire "Braces" (Rudy De Luca), the silver-braced man who organized Dr. Ashley's and Wentworth's murders, to impersonate Thorndyke and shoot a man in the lobby. Now with the police after him, he must prove his innocence. After he is briefly attacked by pigeons, he contacts Brophy, and realizes Brophy took a picture of the shooting. The real Thorndyke was in the elevator at the time, so he should be in the picture.
He orders Brophy to enlarge the picture. When he goes to call, "Braces" tries to strangle him; however, Thorndyke is able to kill him. Brophy enlarges the photo, and Thorndyke is indeed visible in the picture. Nurse Diesel and Montague capture Brophy and take him to the North Wing. They also take the real Arthur Brisbane (Albert Whitlock) to a tower to kill him.
As Thorndyke runs up the tower to save him and Brisbane, Nurse Diesel leaps out from the shadows in a witch's costume with a broom, and falls out the tower window. Thinking she really is a witch, she tries to act like she's flying, ending in her death at the rocks below.
Dr. Montague appears from the shadows and gives up before being hit in the head by the trap door by Brophy. Victoria is reunited with her father, gets married to Thorndyke, and they go off on their honeymoon.

Question: Who did Thorndyke, Diesel and Montatue hire?
Answer: Braces