Watching the movies, I pay a lot of attention to the darkness below the Bridge of Khazad-dûm and the Erebor mines. My question is this:
How far down do the mines go? Is there any information as to how far the Dwarves went before stopping?
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I searched through The Letters of J.R.R. Tolkien, but didn't find anything, and in The History of Middle-Earth, I only found the following brief passage, speaking of the mining of mithril (The Treason of Isengard, p. 185):
'The dwarves tell no tale, but even as mithril was the foundation of their wealth so also it was their destruction: they delved too greedily and too deep, and disturbed that from which they fled.'
A final draft for this passage ends illegibly: 'The dwarves will not say what happened; but mithril is rich only far down and northward towards the roots of Caradras, ...'
And also this short passage, which references the abyss's depth moreso than Moria proper (The Treason of Isengard, p. 431):
'...Little had I guessed the abyss that was spanned by Durin's Bridge.'
'Did you not?' said Gimli. 'I could have told you had there been time. No plummet ever found the bottom - indeed none that was ever case therein was ever recovered.'[*]
What little information I have is from The Two Towers, pg. 491 (emphasis mine):
'Long I fell, and he fell with me. His fire was about me. I was burned. Then we plunged into the deep water and all was dark. Cold it was as the tide of death: almost it froze my heart.'
'Deep is the abyss that is spanned by Durin's Bridge, and none has measured it,' said Gimli.
'Yet it has a bottom, beyond light and knowledge,' said Gandalf. 'Thither I came at last, to the uttermost foundations of stone. He was with me still. His fire was quenched, but now he was a thing of slime, stronger than a strangling snake. We fought far under the living earth, where time is not counted. Ever he clutched me, and ever I hewed him, till at last he fled into dark tunnels. They were not made by Durin's folk, Gimli son of Gloin. Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things....In that despair my enemy was my only hope, and I pursued him, clutching at his heel. Thus he brought me back at last to the secret ways of Khazad-dum: too well he knew them all. Ever up now we went, until we came to the Endless Stair.'
'Long has that been lost,' said Gimli. 'Many have said that it was never made save in legend, but others say that it was destroyed.'
'It was made, and it had not been destroyed,' said Gandalf. 'From the lowest dungeon to the highest peak it climbed, ascending in unbroken spiral in many thousand steps, until it issued at last in Durin's Tower carved in the living rock of Zirakzigil, the pinnacle of the Silvertine.
From this, we see the endless stair seems to stretch the entire depths of Moria, and above it, up to the mountain-top, in many thousand steps. How many is "many thousand"? We don't have that answer, but it seems like that is the best available. We also know that mithril was rich deep down towards the roots of the mountain, so this tells us the mines stretched at least the height of the mountain, and depending on what "roots of the mountain" actually means, could mean miles below the surface. Given that in modern times, gold mines can reach over 2 miles below the surface, it is not improbable to think of Moria being thousands of feet below the base of the mountain, and many more thousands overall.
As far as the Lonely Mountain's mines, there is even less about them. We know that (from The Hobbit):
They buried Thorin deep beneath the Mountain, and Bard laid the Arkenstone upon his breast.
Upon his tomb the Elvenking then laid Orcist, the elvish sword that had been taken from Thorin in captivity. It is said in songs that it gleamed ever in the dark if foes approached, and the fortress of the dwarves could not be taken by surprise.
[*]Footnotes indicate that in earlier drafts the abyss was conceived of as merely a moat, not very deep, but with water that led down into the Deeps
In the film of the Two Towers we see Gandalf and the Balrog in free fall. Terminal velocity for a free falling human is between 50 and 70 meters per second (depending on a lot of things). Early in that sequence we see Gandalf diving head first to catch up with the Balrog, so he was probably falling at the higher end of that range. The Balrog's terminal velocity ought to be higher, but on the other hand it has those bat-like wings and unlike Gandalf its probably trying to slow down. So lets say that its about the same.
The fall sequence lasts 1 minute 12 seconds. At 70 meters per second thats 5040 meters. I'm ignoring the time taken to get to terminal velocity because it is swamped by the uncertainty in terminal velocity.
So it seems that Khazad Dum is around 5 kilometers deep from the Bridge down to the underground lake.
As far as The Hobbit movies are concerned, we have an indirect information about the depth reached by the mines of Erebor:
[Dan Hennah - production designer]: The whole thing about Erebor is it probably goes down as far as the mountain goes up, and they've worked their way right down to the very heart of the mountain and they found the Arkenstone.
BOTFA EE > Appendices part 12 > Realms of the Third Age > Erebor, the Lonely Mountain (at about 20:44)
In the movies the Lonely Mountain is a digital creation modelled on Mount Cook (3,724 m; pictures of the real mountain were purposefully not used out of respect for the Maori culture), so it could be deduced that it is about 3,500 meters high, and the mines go down for a similar extent.