Don't forget the Ottoman Empire, who claimed the title of the Roman Emperor (see Mehmet II), the Tsarist Russia, who claimed the title of the Emperor of Rome, and the Holy Roman Empire, since the Pope crowned Charmalange the Roman Emperor.
So, really, the Roman Empire lasted until the 1920's, when Turkey abolished the Sultanate and thus ended the Ottoman Empire.
Actually, come to think of it, would you count His Excellency Benito Mussolini, Head of Government, Duce of Fascism, and Founder of the Empire as a de facto Roman Emperor? His supporters called the Italian colonial empire as the New Roman Empire after all. If so, the Roman Empire finally collasped in 1945 when Mussolini was killed by partisans.
EDIT: Come to think of it. According to Wikipedia, once Odoacer overthrew Romanalus Augustus, he sent a letter to Zeno, Emperor of the Eastern Roman Empire, asking Zeno to recognize him Patricius of the Roman Empire and Zeno's viceroy in Italy. Zeno accepted, on the condition Odoacer declare Flavius Julius Nepos as the Western Roman Empire. Which Odoacer did, because Flavius Julius Nepos was nothing more than a puppet. Once Flavius Julius Nepos died, Odoacer called himself the King of Italy...altough, if Odoacer decided to call himself the Western Roman Emperor, prehaps the Western Roman Empire may still exist...
EDIT2: Since Augustus took the title Emperor in January 16 27 BCE...I say the Roman Empire has existed for, say, 1918 years.
And I think the reason people date the Republic as part of the Empire is becasue only Romans were allowed to vote, and Italians only granted the right to vote very late in the Republic, during the Social War. Therefore, the Roman Republic had an colonial empire, hence why people say it lasted for over 2000 years.