Karen Wynn Fonstad's highly regarded The Atlas of Middle-earth includes a diagram showing a vertical cross section of Moria.
In the diagram, the horizontal distance of 20 miles from the guardroom to the bridge appears that to be about 2.5 times the vertical drop from the "First Deep" to the bottom of the abyss. Accounting for the 7:1 exaggeration of height in the diagram, that makes the drop about (20 / 2.5 / 7 =) 1.14 miles.
While The Atlas of Middle-earth provides several references to the text of The Lord of the Rings, I can't find any that back up the height of the vertical drop to the abyss that is shown in the diagram. It does, however, provide a reference to back up the 40 miles from West-door to East-gate (20 miles from the guardroom to both the West-door and East-gate).
‘I cannot say,’ answered Gandalf. ‘It depends on many chances. But going straight, without mishap or losing our way, we shall take three or four marches, I expect. It cannot be less than forty miles from West-door to East-gate in a direct line, and the road may wind much.’
The Lord of the Rings Book Two, Chapter 4: A Journey in the Dark
Page 310 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
