In Tolkien’s own words
I have long wished to see a full account of the War of the Last Alliance as written by Tolkien for the history of Middle-earth. Unfortunately, much of what we know is scattered into fragments over numerous books.
Following the success of The Tale of the Dagor Dagorth post, I have attempted to do the same thing. Looking for every passage, sentence, footnote and scrap of information referencing the Last Alliance, I have tried to construct a full account using Tolkien’s own writings.
The use of square brackets  indicates a slight change from my end to connect the different sentences together, whilst the numbers refer to the books from which they are extracted.
I hope you find the following account as entertaining as I have had compiling it.
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.
The Old Forest Road that led down from the Pass of Imladris and crossed Anduin by a bridge […] had been enlarged and strengthened for the passage of the armies of the Alliance.
Sauron, well-informed of the Alliance, […] sent out such Orc-troops of the Red Eye as he could spare, to do what they could to harry any forces that attempted to shorten their road by crossing the Mountains. In the event the main might of Gil-galad, together with Isildur and part of the Men of Arnor, had come over the Passes of Imladris and Caradhras, and the Orcs were dismayed and hid themselves.
From Imladris they crossed the Misty Mountains by many passes and marched down the River Anduin. [Isildur] journeyed […] up the Vales of Anduin to Cirith Forn en Andrath, the high-climbing pass of the North, that led down to Imladris […] and marched that way to the war with men of eastern Arnor in the company of Elrond[;] and so came at last upon the host of Sauron on Dagorlad, the Battle Plain, which lies before the gate of the Black Land.
With Isildur went his three sons, Elendur, Aratan, and Ciryon. All three had fought in the War of the Last Alliance, but Aratan and Ciryon had not been in the invasion of Mordor and the siege of Barad-dur, for Isildur had sent them to man his fortress of Minas Ithil, lest Sauron should escape Gil-galad and Elendil.
Amdir [father of Amroth] obeyed the summons of Gil-Galad and brought as large a force as he could muster to the Last Alliance.
Oropher [the father of Thranduil, father of Legolas] had the wisdom to foresee that peace would not return unless Sauron was overcome. He therefore assembled a great army of his now numerous people, and joining with the lesser army of Amdir of Lorien he led the host of the Silvan Elves to battle.
All living things were divided in that day, and some of every kind, even of beasts and birds, were found in either host, save the Elves only. They alone were undivided and followed Gil-galad.
The Silvan Elves were hardy and valiant, but ill-equipped with armour or weapons in comparison with the Eldar of the West; also they were independent, and not disposed to place themselves under the supreme command of Gil-galad. Their losses were thus more grievous than they need have been, even in that terrible war. Amdir and more than half his following perished in the great battle of the Dagorlad, being cut off from the main host and driven into the Dead Marshes. Oropher was slain in the first assault upon Mordor, rushing forward at the head of his most doughty warriors before Gil-galad had given the signal for advance.
Of the Dwarves few fought upon either side; but the kindred of Durin of Moria fought against Sauron.
[Elrond] was the herald of Gil-galad and marched with his host. He was at the Battle of Dagorlad before the Black Gate of Mordor, where [the Elves] had the mastery: for the Spear of Gil-galad and the Sword of Elendil, Aiglos and Narsil, none could withstand.
The host of Gil-galad and Elendil had the victory, for the might of the Elves was still great in those days, and the Númenóreans were strong and tall, and terrible in their wrath. The Númenóreans in their own land possessed horses, which they esteemed. In the War of the Last Alliance such horses as they used had suffered great losses.
Then Gil-galad and Elendil passed into Mordor and encompassed the stronghold of Sauron; and they laid siege to it for seven years, and suffered grievous loss by fire and by the darts and bolts of the Enemy, and Sauron sent many sorties against them. There in the valley of Gorgoroth Anárion son of Elendil was slain, and many others. [The] helm of Anárion was crushed by the stone-cast from Barad-dûr that slew him.
But at the last the siege was so strait that Sauron himself came forth; and he wrestled with Gil-galad and Elendil. [Elrond] beheld the last combat on the slopes of Orodruin [… Isildur] alone stood by his father in that last mortal contest; and by Gil-galad only Círdan stood, and Elrond. [With] the heat of Sauron’s hand, which was black and yet burned like fire, Gil-galad was destroyed. But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places; and he took no visible shape again for many long years.
The Dark Tower was broken, but its foundations were not removed; for they were made with the power of the Ring and while it remains they will endure. Many Elves and many mighty Men, and many of their friends had perished in the war […] Never again shall there by any such league of Elves and Men.
Timeline of the War
3430: The Last Alliance of Elves and Men is formed.
3431: Gil-galad and Elendil march east to Imladris.
3434: The host of the Alliance crosses the Misty Mountains. Battle of Dagorlad and defeat of Sauron. Siege of Barad-dûr begins.
3440: Anárion slain.
3441: Sauron overthrown by Elendil and Gil-galad, who perish. Isildur takes the One Ring.