Kirkman and his wife are at a secure location (which just happens to overlook Washington), eating popcorn and watching the president's speech. When the broadcast abruptly cuts off, Secret Service agents rush into the room as news reports of an explosion come on the air. Kirkman flings open the blast-door shutters just in time to see a fireball plume over the Capitol, and it's clear that he's in Rosslyn. The real designated survivors are nowhere near the nation's capital.
Kirkman doesn't find out that he's been selected to sit out the speech until he's preparing to leave for the Capitol. His cellphone rings. "What's a designated survivor?" he asks. In reality, the selected official would know days before and would be secretly whisked out of Washington on the day of the address.
The "designated survivor" is not held in a location within sight of the Capitol building as portrayed. Instead, the person is taken to Mount Weather, VA, which is almost 50 miles from Washington, DC.
Both the Senate and the House establish a designated survivor. Each federal agency has a continuity of government plan that determines their line of succession. So, in this event, there would be an acting president, Speaker of the House, president pro tempore of the Senate, and Cabinet.
At the end of the program, President Kirkman requests a meeting with Iran's ambassador to the U.S. According to the U.S. State Department Web site, the U.S. does not currently have diplomatic relations with Iran.