What are the examples of deus ex machina in the last act of The Man Who Came to Dinner?

The literal definition of “Deus ex Machina” is “God from the machine.” Originally the phrase referred to a device used in ancient Greek theater, in which actors playing deities were maneuvered onto the stage in order to bless or condemn the mortal characters - or to provide an otherwise unattainable resolution. However, the term has evolved over the years away from an actual machine to merely a plot device denoting any unexpected development that neatly provides a solution to an insurmountable narrative difficulty.
In the case of “The Man Who Came to Dinner,” (written by Moss Hart & George S. Kaufman) we see two uses of a Deus Ex Machina. First is the arrival of the mummy case at just the point where Sheridan Whiteside is desperate to remove Lorraine Sheldon from the area. Though previous acts have shown Whiteside receiving all sorts of strange gifts from his varied and numerous friends around the globe, the fact that this particular one shows up exactly when needed fulfills the definition to a nicety.
The second appearance of a Deus ex Machina is Whiteside’s resurfaced memory that Mr. Stanley’s sister is not Harriet Stanley, but Harriet Sedley, an infamous ax murderess. This knowledge allows Whiteside to blackmail Stanley into resolving his daughter’s potential marriage and his son’s career to their (and Whiteside’s) satisfaction, as well as preventing himself being thrown out of the house by an exasperated Mr. Stanley. While hints of this revelation were planted in earlier acts, the timing of the revelation - again, just when needed to resolve things - perform the Deus ex Machina function perfectly.

There are three examples of deus ex machina in the third act of The Man Who Came to Dinner, namely, the arrival of Banjo, the appearance of a mummy case, and Sherry recognizing a photograph.
Deus ex machina is a literary device that allows a person or item to solve problems that might otherwise go unsolved. It is usually contrived and improbable.
Three instances of deus ex machina appear in the last act of The Man Who Came to Dinner and work together to resolve the problems of the characters in the play. Without the occurrence of these improbable events, things would not have worked out for Sheridan, who wants to remove Lorraine from Mesalia.
The first instance is the arrival of Banjo, a friend of Sherry who works as a comedian. He happens to have a plane.
The second instance is the appearance of a mummy case that is delivered to Sherry as a Christmas gift from the Khedive of Egypt. It provides a hiding place for the characters to smuggle Lorraine out.
The last instance is Sherry recognizing a photograph of his host's aunt, a well-known murderer. This allows him to blackmail his host into helping them get the mummy case to Banjo's plane and remove Lorraine from Mesalia. 
George S. Kaufman and Moss Hart use three instances of deus ex machina to resolve the action of their play in the final act. It would be a happy ending for the main character, if he did not slip on the ice on his way to the train station, condemning him to another period of convalescence. 

A deus ex machina is a plot device, usually somewhat contrived, that brings about a resolution in a drama. There are two examples in the last act of The Man who Came to Dinner. The first is when a large mummy case is delivered to Sherry from the Khedive of Egypt while Sherry is staying at the house of the Stanley family. Banjo and Sherry lock Lorraine, who has been flirting with Bert, in the mummy case. With Lorraine out of the way, Bert can again focus his energies on Maggie, who really loves him. This is an example of a deus ex machina because it's a contrived means of removing Lorraine so that Maggie and Bert can be together. 
In addition, Sherry notices a photo of Harriet Stanley, the eccentric old aunt in the house, when she is younger. He recognizes her face as that of a murderer, and he uses this information to force Mr. Stanley to help him get the mummy case onto a plane. This is another example of a deus ex machina as it's a plot element that comes out of nowhere to help Sherry achieve his goal of putting Lorraine on a plane. Finally, Sherry slips again on the ice as he is finally leaving the Stanley house and is brought back into the house--another example of a deus ex machina. This final plot device is a somewhat contrived end to the play, as it explains what will happen to Sherry after the play ends (he'll have to stay on at the house for the immediate future).