According to The Annals of Aman in volume X of The History of Middle-earth, the Lamps of the Valar fell in Valian Year 3450, then the Two Trees were made 50 Valian Years later, and stood for 1,495 Valian Years. Converting these numbers into solar years is complicated by the fact that Tolkien experimented with various formulations and calculations, but he appears - at least provisionally - to have settled on a figure of 33,530 years from the making of Arda to the fall of the Lamps, and a total of 50,775 from the beginning of the world to the first rising of the Sun.
These numbers give a period of 17,245 between the making of the Two Trees and the rising of the Sun, to which we need to add a further 479 (based on the fifty Valian Years between the fall of the Lamps and the first Light of the Trees), giving a total length for the Long Night of 17,724 years.
Although this might appear to be a remarkably exact figure, it should be emphasised that it is only one possible value from several dating schemes that Tolkien considered. The alternative schemes all use relatively similar values for solar years (within about a thousand years of one another) and so, while this period of 17,724 may not be precisely correct, it certainly gives a realistic idea of the number of millennia that Middle-earth lay under the starlit darkness of the Long Night.
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