A complete tale of the War of the Last Alliance is a tantalising prospect. Yet, all the information we have available comes from fragments scattered over a number of books. The following post attempts to reconstruct the entire account as written Tolkien himself
This was made possible by searching for every passage, sentence, footnote and scrap of information referencing the Last Alliance.
The use of square brackets  indicates a slight rewording to connect the different sentences together, whilst the numbers refer to the books from which they are extracted.
The Tale of the War of the Last Alliance
Then Elendil the Tall and his mighty sons, Isildur and Anarion, became great lords; and the North-realm they made in Arnor and the South realm in Gondor above the mouths of Anduin. But Sauron of Mordor assailed them.
Now Elendil and Gil-galad took counsel together, for they perceived that Sauron would grow too strong and would overcome all his enemies one by one, if they did not unite against him. Therefore they made that League which is called the Last Alliance. It is told that Elendil stood [on Weathertop] watching for the coming of Gil-galad out of the West and the hosts of Gil-galad and Elendil were mustered in Arnor. [A]nd they marched east into Middle-earth gathering a great host of Elves and Men; and they halted for a while at Imladris. It is said that the host that was there assembled was fairer and more splendid in arms than any that has since been seen in Middle-earth. It recalled the glory of the Elder Days and the hosts of Beleriand, so many great princes and captains were assembled, and none greater has been mustered since the host of the Valar went against Thangorodrim.