Author's Notes

Tolkien doesn't have a whole lot to say about the Battle of Dale or the events leading up to it, but there's enough canon to get a sense of the timeline and general troop movements as well as make an educated guess as to the sizes of the armies involved. First, the borders of Brand's kingdom have to be drawn.

[Glóin:] 'Nowhere are there any men so friendly to us as the Men of Dale. They are good folk, the Bardings. The grandson of Bard the Bowman rules them, Brand son of Bain son of Bard. He is a strong king, and his realm now reaches far south and east of Esgaroth.' (The Lord of the Rings: The Fellowship of the Ring, Book II, Chapter I, "Many Meetings")

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To the Great Hall of Thráin, Thrór brought back the Arkenstone, and he and his folk prospered and became rich, and they had the friendship of all Men that dwelt near. For they made not only things of wonder and beauty but weapons and armour of great worth; and there was great traffic of ore between them and their kin in the Iron Hills. Thus the Northmen who lived between Celduin (River Running) and Carnen (Redwater) became strong and drove back all enemies from the East; and the Dwarves lived in plenty, and there was feasting and song in the Halls of Erebor. (The Lord of the Rings, Appendix A, "Durin's Folk")

Assuming then that the Northmen are as strong or more so, united under Brand's crown (Esgaroth excluded, perhaps), as when Thrór ruled as King Under the Mountain, the greater realm of Dale likely consists of all the land between the River Running in the west and the Redwater in the east, which takes its name from the ore in the Iron Hills. This is an area roughly comparable to Rohan, and the Bardings, who are kin to the Rohirrim, probably live in much the same way, in scattered farming and fishing villages. Dale is, of course, the largest city but, strategically, I'd expect there to be another fortified town where the Celduin and Carnen meet at the kingdom's southernmost point.

In military terms and, again, using Rohan as a model, I feel an army of no more than 10,000 is plausible for the whole of Dale, with the town of Dale supplying a good half of this force—not only infantry and cavalry, but a small fleet to patrol and defend the rivers. Brand's problem, like Théoden's on the eve of the Battle of the Pelennor Fields, is that his is not primarily a standing army and mustering soldiers who are usually busy being farmers and fishermen takes time on the order of weeks. The Easterlings are not so accomodating as that. When the Council of Elrond convenes on October 25 (The Lord of the Rings, Appendix B: The Tale of Years), they're already posturing threateningly on their side of the Redwater.

[Glóin:] 'We discover that messengers [from Mordor] have come also to King Brand in Dale and that he is afraid. We fear that he may yield. Already war is gathering on his eastern borders. If we make no answer, the Enemy may move Men of his rule to assail King Brand and Dáin also.' (The Lord of the Rings: The Fellowship of the Ring, Book II, Chapter II, "The Council of Elrond")

From Appendix B, some relevant dates:

March

11 | First assault on Lórien.
14 | Minas Tirith is besieged.
15 | Battle under the trees in Mirkwood; Thranduil repels the forces of Dol Guldur. Second assault on Lórien.
17 | Battle of Dale. King Brand and King Dáin Ironfoot fall. Many Dwarves and Men take refuge in Erebor and are besieged.
22 | Third assault on Lórien.
25 | Downfall of Barad-dûr and passing of Sauron.
27 | Bard II and Thorin III Stonehelm drive the enemy from Dale.
28 | Celeborn crosses Anduin; destruction of Dol Guldur begun.

April

6 | Meeting of Celeborn and Thranduil.

Lastly, the account of the battle and siege, also from Appendix B:

At the same time as the great armies besieged Minas Tirith, a host of the allies of Sauron that had long threatened the borders of King Brand crossed the River Carnen, and Brand was driven back to Dale. There he had the aid of the Dwarves of Erebor; and there was a great battle at the Mountain's feet. It lasted three days, but in the end both King Brand and King Dáin Ironfoot were slain, and the Easterlings had the victory. But they could not take the gate, and many, both Dwarves and Men, took refuge in Erebor and there withstood a siege.

When news came of the great victories in the South, then Sauron's northern army was filled with dismay; and the besieged came forth and routed them, and the remnant fled into the East and troubled Dale no more. Then Bard II, Brand's son, became King in Dale, and Thorin III Stonehelm, Dáin's son, became King Under the Mountain.

Note that the Easterlings cross the Carnen on March 14, but the battle at the Mountain's feet doesn't begin until three days later and then lasts three days in what, if I'm not mistaken, is the longest single engagement during the War of the Ring. My guess is that, owing to the difficulties of mustering his forces quickly and defending the entire length of his eastern border, Brand doesn't have the numbers to do more than harry the enemy after he fails to stop them at the river. Once he retreats to Dale, he's joined by Dáin and probably several thousand Dwarven warriors, and together with the advantages of better weapons and armor, whatever fortifications had been raised, they slow the enemy to a crawl, such that it takes the Easterlings three bloody days to press down the valley between the Mountain's eastern and southern spurs, as described in The Hobbit, to reach the gates of Erebor. That is where Brand and Dáin finally fall, as night gathers on March 19/20.

While the Easterlings have the victory in the Battle of Dale, it costs them so severely that they can only besiege the Mountain, for a full week, rather than overwhelming its defenses by storm. Not to mention, they almost kind of give up and go home or the military equivalent. Besides low morale upon hearing word of Sauron's fall (the Eagles!), I've speculated that the Easterlings suffer from a lack of command unity—possibly because they were intended to join the forces of Dol Guldur or vice versa with the three Nazgûl, led by Khamûl, himself once an Easterling, originally in command there as their generals; possibly because their best fighting units were deployed in the south against Gondor—and are attacked by levies from Dale's outlying villages come to break the siege. Helm's Deep and Minas Tirith are similarly relieved by the defenders sallying forth when new reinforcements arrive on the field.

Here Smaug routs the Easterlings on the morning of March 12, leaving Brand and Dáin time enough to gather their forces and march to Thranduil's aid in Mirkwood. The Battle of Dale never happens, and neither does the third assault on Lórien. Instead, with the Men and Dwarves lending his enemies unexpected numbers, Sauron's orc army is not only defeated in short order but driven south to Dol Guldur, which is besieged and destroyed before the Ring is cast into the fires of Mount Doom. News of victory in the north is brought by the Eagles to the Captains of the West at the Black Gate, heartening the allied host for the final effort. Emissaries from Dale and Erebor, with a strange tale to tell, follow down the Anduin for King Elessar's coronation on May 1.