The basic answer to your question of whether he used archaic language to emulate the Bible is no, of course not. There is one specific exception, however, which I explain further on.
The Lord of the Rings is composed entirely in Modern English using all manner of style and register. This is deliberate. (I will exempt Éomer’s “Westu Théoden hál!” taken directly from Beowulf’s “Wæs þú, Hróðgár, hál”, as that actually is Old English. Nothing else is.)
Tolkien himself specifically explains exactly why he resorts to the second-person singular in certain situations. Please read Appendix F, section II, “On Translation” for this and a great deal more about what he was doing, and why, when he attributed different styles of speech to different characters at different times. The Witch-king’s use of the second person was intended to be belittling.
In Tolkien’s letter to Milton Waldman, while attempting to present a summary of the book’s plot and character, Tolkien states that:
. . . even in style it is to include the colloquialism and vulgarity of the Hobbits, poetry and the highest style of prose.
So his mixing of low and high, and even of poetry and prose, was done with full awareness and intent. Indeed, in the same Waldman letter he also wrote:
Hardly a word in its 600,000 or more has been unconsidered. And the placing, size, style, and contribution to the whole of all the features, incidents, and chapters has been laboriously pondered
Perhaps the best explanation for why Tolkien ever used archaic phrasing can be found in his Letter #171, wherein he takes up the accusation of gratuitous “tushery” (archaism). He addresses the point of why one would use archaic language to represent what was (purported to be) spoken by an archaic people in this way.
....Don’t be disturbed! I have not noticed any impertinence (or sycophancy)
in your letters; and anyone so appreciative and so perceptive is
entitled to criticism. Anyway I do not naturally breathe an air of
undiluted incense! It was not what you said (last letter but one,
not the one that I answered) or your right to say it, that might
have called for a reply, if I had the time for it; but the pain
that I always feel when anyone – in an age in which almost all
auctorial manhandling of English is permitted (especially if
disruptive) in the name of art or ‘personal expression’ – immediately
dismisses out of court deliberate ‘archaism’. The proper use of
‘tushery’ is to apply it to the kind of bogus ‘medieval’ stuff which
attempts (without knowledge) to give a supposed temporal colour
with expletives, such as tush, pish, zounds, marry, and the like.
But a real archaic English is far more terse than modern; also many
of things said could not be said in our slack and often frivolous
idiom. Of course, not being specially well read in modern English,
and far more familiar with works in the ancient and ‘middle’ idioms,
my own ear is to some extent affected; so that though I could easily
recollect how a modern would put this or that, what comes easiest
to mind or pen is not quite that. But take an example from the
chapter that you specially singled out (and called terrible): Book
iii, ‘The King of the Golden Hall’. ‘Nay, Gandalf!’ said the King.
‘You do not know your own skill in healing. It shall not be so. I
myself will go to war, to fall in the front of battle, if it must
be. Thus shall I sleep better.’
This is a fair example – moderated or watered archaism. Using only
words that still are used or known to the educated, the King would
really have said ‘Nay, thou (n’)wost not thine own skill in healing.
It shall not be so. I myself will go to war, to fall...’ etc. I
know well enough what a modern would say. ‘Not at all, my dear G.
You don’t know your own skill as a doctor. Things aren’t going to
be like that. I shall go to the war in person, even if I have to
be one of the first casualties’ – and then what? Theoden would
certainly think, and probably say ‘thus shall I sleep better’! But
people who think like that just do not talk a modern idiom. You can
have ‘I shall lie easier in my grave’, or ‘I should sleep sounder
in my grave like that rather than if I stayed at home’ – if you
like. But there would be an insincerity of thought, a disunion of
word and meaning. For a King who spoke in a modern style would not
really think in such terms at all, and any reference to sleeping
quietly in the grave would be a deliberate archaism of expression
on his part (however worded) far more bogus than the actual ‘archaic’
English that I have used. Like some non-Christian making a reference
to some Christian belief which did not in fact move him at all.
Or p.127, as an example of ‘archaism’ that cannot be defended as
‘dramatic’, since it is not in dialogue, but the author’s description
of the arming of the guests – which seemed specially to upset you.
But such ‘heroic’ scenes do not occur in a modern setting to which
a modern idiom belongs. Why deliberately ignore, refuse to use the
wealth of English which leaves us a choice of styles – without any
possibility of unintelligibility. I can see no more reason for not
using the much terser and more vivid ancient style than for changing
the obsolete weapons, helms, shields and hauberks into modern
‘Helms too they chose’ is archaic. Some (wrongly) class it as
‘inversion’, since normal order is ‘They also chose helmets’ or
‘they chose helmets too’. (Real mod. E. ‘They also picked out some
helmets and round shields.) But this is not normal order, and if
mod. E. has lost the trick of putting a word desired to emphasize
(for pictorial, emotional or logical reasons) into prominent first
place, without addition of a lot of little ’empty’ words (as the
Chinese say), so much the worse for it. And so much the better for
it the sooner it learns the trick again. And some one must begin
the teaching, by example.
I am sorry to find you so affected by the extraordinary 20th C.
delusion that its usages per se and simply as ‘contemporary’ –
irrespective of whether they are terser, more vivid (or even nobler!)
– have some peculiar validity, above those of other times, so that
not to use them (even when quite unsuitable in tone) is a solecism,
a gaffe, a thing at which one’s friends shudder or feel hot under
the collar. Shake yourself out of this parochialism of time! Also
(not to be too donnish) learn to discriminate between the bogus and
genuine antique – as you would if you hoped not to be cheated by a dealer!
He does not actually write archaic language, or not fully archaic, for he
would have lost the reader. What he does do at times is watered or moderated archaism, and he does this for a distinct reason given above.
Every linguistic effect he used has a specific purpose that makes good sense in that context. Tolkien was a master of this, and if you think it all archaic, you haven’t been reading closely enough. Look at the rustic speech of the hobbits or the coarse speech of the orcs and trolls: all styles and registers are represented in the book, from the low to the very highest.
I would say that most uses of archaic language in fantasy literature written today are fake. That’s because there is so much of it, and Sturgeon’s Law applies. They are the very embodiment of what Tolkien complained of in his letter to Hugh Brogan. He himself doesn’t do that — because he actually knew what he was writing. Most people do not, and so their would-be antique language comes off sounding bogus, not authentic.
About that exception.
The one, and I believe only, place where Tolkien deliberately echoed the KJV is in the chapter “The Stewart and the King” from Book VI. The eagle’s song found there is quite obviously cast in the stylized language of the Psalms. I believe Tolkien felt that nothing short of Biblical language could suitably convey the “sudden joy as poignant as grief” of the unexpected eucatastrophe after all but the wannest of hopes had been forgotten.
You really should read Tom Shippey’s The Road to Middle-Earth: How J. R. R. Tolkien Created a New Mythology, preferably the 2003 edition. Shippey discusses Tolkien’s use of language at length. He also shows that all modes of Frye’s Anatomy of Criticism are present based on the characters involved, all the way from the mythic to the ironic.