Actually, I think the replies above are right on the money.
I am guessing the balrog was asleep down in the deeps, and Balin's (foolish) expedition failed to arouse it.
Balin wasn't so foolish he sent dwarves down into the really deep places, as he reclaimed Moria, or there would have been no book for Gandalf to read from.
And I can well believe some of the orcs of Moria (there weren't many, due to the Battle of Five Armies) survived the dwarven onslaught, and fled into the deeps, and waited.
I am guessing that Dul Guldur was the staging around for the counterattack.
Going around Lothlorien and crossing Anduin, the orc host came up the Silverlode into Dimril Dale, where their scouts caught Balin alone and slew him.
The main host arrived, only to find the way blocked by the East Gates of Moria, which the dwarves had barred.
However, there were far more orcs, apparently, than there were dwarves, and the siege was on.
The dwarves apparently sent Oin to see if an avenue of retreat was possible through West Gate, and the survivors of that expedition came back to inform the dwarves that no such avenue existed, and that Oin had been killed.
Meanwhile, the orcs beat their way through attrition through the Gates.
Once more, they were halted - at the Bridge of Khazad-Dum.
Once more, the sheer numbers of orcs wore down the defenders (I assume the orcs paid greatly for the Bridge), and once more the dwarves fell back.
In a bloodbath of a battle, the orcs took the Second Hall, and the dwarves moved their command center to the Chamber of Mazarbul, where they were now effectively trapped.
Thence, the orcs assembled for one great, final assault on the dwarves, even as Ori penned the fateful words in his book: we cannot get out.
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It is a pity the elves of Lothlorien were estranged from the dwarves.
An assault by the elves of Lothlorien, at the crossings of Anduin, could have stopped the orc host dead in it's tracks.
However, short of Galadriel herself going into Khazad-Dum (and maybe not even then) there is no way Balin could have hoped to win against what he called Durin's Bane.
And he knew Durin's Bane was in there, somewhere. King Dain would have told him so.
What were Balin, Ori, and Oin thinking of??
Lord of Moria? A bit pretentious, don't you think?
A proclamation made while the balrog was still the true Lord of Moria (as Colonel Hardisson might have put it, a person does not proclaim himself mayor of Los Angeles when one is only mayor of Palm Springs far east of L.A.)
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By the way, somewhere it was mentioned that we might see more of the battle between Gandalf and the balrog in the film The Two Towers.
I hope so.
That would be great (even if it would be very sad, for remember - although the balrog dies, it mortally wounds Gandalf too. In other words, the balrog kills Gandalf.)