Factual error: The show depicts the vote ending in a 50-50 tie, and then shows VP Doyle casting a tie-breaking vote for Montez (which the show then erroneously treats as Montez being elected president - she would simply be the VP "acting as president" for as long as the House fails to resolve the stalemate). 67 senators have to cast votes for the meeting to count, and 51 votes are needed to be VP. A 50-50 tie in this case. The 12th amendment actually provides a procedural exception to the tie-breaking power, by stating that the majority of whole Senators, in this case, 51, are necessary for the selection of the VP. As the vice president is not a senator, his vote would not have an effect on reaching the necessary 51, and thus a 50-50 vote would simply trigger a new ballot, and the senate would continue to vote until someone is elected. In this regard, the show makes another mistake with an on-screen graphic identifying Doyle as a senator, and not the vice president, who while given the constitutional role of president of the senate is not actually a senator.
Plot hole: In season five, the show depicts the aftermath of an Electoral College tie. The procedure in this case should be the House holding subsequent ballots until a president is elected. On the show, however, Tom James convinces the Speaker of the House to hold one ballot, and then not vote again. James' plan is to win the Senate vote for VP, then act as president for four years before being elected to two full terms as actual president. James is outmaneuvered and his rival Montez is elected VP, and subsequently acts as president for the remainder of the series. After season 5, the show makes no mention of the House ever taking up a vote for president again, and the show simply treats Montez as the actual president. A Speaker of the House blocking the election of a new president would likely cause a political uprising from supports of both candidates, and both candidates would rightly take to the airwaves to demand a new vote. The idea of a power-hungry politician such as Selina, who uses every trick in the book to promote herself and elevate her own power, putting up no fight is just bizarre.