In the beginning of the episode, Sam and Josh build a fire in the Mural room fireplace. They find the the chimney has been sealed since the 1800's causing the room to fill with smoke causing the smoke alarms to go off. In season 1 in the episode where Mendoza is confirmed, everyone is watching the vote in the Mural room while a fire is burning in the same fireplace.
In "Six Meetings Before Lunch" there is a fire in a fireplace in the Mural Room. Yet in "The Leadership Breakfast" Sam and Josh try to light a fire in the same fireplace which has had the flue welded shut for 100 years.
President Bartlett mentions it is January, but the trees outside of the Oval Office have all of their leaves and it looks all the world like summer outside..
When Sam and Josh start a fire in the Mural Room (which does not exist in the actual White House), Sam discovers a bronze plaque which states that the fireplace flue had been sealed for one 100 years and that the fireplace was a favorite place for President Andrew Johnson. Since the fictional Mural Room is depicted throughout the series as being located in the West Wing of the White House, it could not have existed for 100 years or during the presidency of Andrew Johnson. The West Wing itself did not exist at that time. It was not constructed until the administration of Theodore Roosevelt and was reconstructed entirely twice during the Hoover administration.
Charlie hands President Bartlet some remarks which Sam produced for the breakfast. President Bartlet reads aloud (with approval) from the text: "We spend so much time demonizing the other side, treating our opponents as if they were threatening strangers with whom we share nothing in common." Neither Bartlet nor Sam would have let a redundancy like "share nothing in common" into the text of a Presidential address.