L'instrument le plus cité est, justement, la harpe:
Dans "The Hobbit":
Dans les chansons des Nains:
Goblets they carved for themselves
And harps of gold;

King under mountains:
His harp shall be restrung

Des harpes sont mentionnées dans le trésor de Smaug:
finding still hanging there many golden harps strung with silver they took them and struck them; and being magical (and also untouched by the dragon, who had small interests in music) they were still in tune.
Mais rien n'indique qu'elles soient de fabrication Naine, même si c'est le cas le plus plausible.

Lors du siège du Royaume sous la Montagne, on entend les harpes des Elfes, auxquelles les Nains répondent par les leurs.
There was the sound, too, of elven-harps and of sweet music (...)
Then the Dwarves themselves brought forth harps ans instruments regained from the hoard, and made music to soften his mood; but their song was not as elvish song.

Dans 'The Lord of the Ring":
Chant de Gil-Galad:
Gil-Galad was an Elven-King
Of him the harpers sadly sing

Chant de Gimli en Moria:
The harpers harped, the minstrels sang

Haldir en Lorien:
our hands are more often upon the bowstring than upon the harp.

Galadriel:
in her hand she held a harp, and she sang.

Au couronnement d'Aragorn:
and from Dol Amroth came the harpers that harped most skilfully in all the land; and there were players upon viols and upon flutes and upon horns of silver, and clear-voiced singers from the vales of Lebennin.

and amid the music of harp and of viol and of flute and the singing of clear voices the King passed through the flower-laden streets

Elrond aux Havres:
Elrond wore a mantle of grey and had a star upon his forehead, and a silver harp was in his hand,

Dans "The Silmarillion":
Fingon;
Then in defiance of the Orcs, who cowered still in the dark vaults beneath the earth, he took his harp and sang a song of Valinor that the Noldor made of
old, before strife was born among the sons of Finw‘; and his voice rang in the mournful hollows that had never heard before aught save cries of fear and woe.

La venue des hommes dans l'ouest:
Then he went among the sleeping people, and sat beside their dying fire where none kept watch; and he took up a rude harp which Bëor had laid aside, and he played music upon it such as the ears of Men had not heard; for they had as yet no teachers in the art, save only the Dark Elves in the wild lands.
Now men awoke and listened to Felagund as he harped and sang, and each thought that he was in some fair dream, until he saw that his fellows were awake also beside him; but they did not speak or stir while Felagund still played, because of the beauty of the music and the wonder of the song.

Narn i Hurin
It is told that a seer and harp-player of Brethil named Glirhuin made a song, saying that the Stone of the Hapless should not be defiled by Morgoth nor ever thrown down, not though the sea should drown all the land; as after indeed befell, and still Tol Morwen stands alone in the water beyond the new coasts that were made in the days of the wrath of the Valar.

Elle est donc plus souvent attribuée aux Nains dans "The Hobbit", et plus souvent aux Elfes dans "The Lord of the Ring". Elle est la plupart du temps associée au chant. On notera que lorsque les Nains font de la musique chez Bilbo, c'est Thorin qui en joue. Pour les Elfes, Galadriel et Elrond sont nommés. La harpe semble donc être l'attribut de personnages importants.

Et bien évidemment, elles apparaissent dans l'Ainulindale, en compagnie d'autres instruments:
Then the voices of the Ainur, like unto harps and lutes, and pipes and trumpets, and viols and organs, and like
unto countless choirs singing with words, began to fashion the theme of Iluvatar to a great music;

Les autres instrument cités lorsque les Nains font de la musique chez Bilbo sont:
fiddles (joués par Kili et Fili): le fiddle est le violon, mais joué dans un style particulier, comme dans la musique irlandaise. Il est curieux de le voir en compagnie de violes, puisque le violon en est le descendant.
flutes (jouées par Dori, Nori et Nori). Le terme en anglais désigne plus la flûte traversière que la flûte à bec (recorder).
drum (joué par Bombur). On ignore de quel tambour il s'agit, mais on peut supposer qu'il est d'assez grande taille.
clarinets (jouées pas Bifur et Bofur). La présence de cet instrument est plutôt étrange. Comme pour le violon, on peut soit penser qu'il s'agit d'une de ces incongruités propres à "The Hobbit" ou voir là une preuve de l'ingéniosité des Nains qui ont réussi à développer des instruments complexes à fabriquer.
viols (jouées par Dwalin et Balin).

Et enfin, lors de l'anniversaire de Bilbo:
Noises of trumpets and horns, pipes and flutes, and other musical instruments.
Le terme de "pipes" est ambigu, car il désigne soit un tuyau, et donc une flûte, soit plus spécifiquement le bagpipe, c'est à dire la cornemuse.

Pour la flûte, je me souviens vaguement d'un poème à propos de Tifang Warble dans "The Book of lost Tales" mais je n'ai pas le livre sous la main. Si quelqu'un pouvait préciser ça, En tout cas, il s'agissait d'un Elfe, et plutôt expert en la matière.