What happened to the 13 Rings of Power that survived the War of the Ring? I'm talking about the 9 given to the kings of Men and the 7 rings given to the Dwarves (3 destroyed by dragons, 4 recovered by Sauron). Sauron had a large amount of Mithril as well, what happened to it? He was said to covet Mithril above all else, so maybe he would keep the Rings of Power and Mithril together somewhere?
It is unclear what happened to the Rings held by Sauron or the Mithril he coveted.
The books never go into extensive details (neither do the supplemental materials) of the outcome of the additional rings or what Sauron had intended to do with the mithril beyond its use for weapon making. We do, however, get a few details about the three Elven rings, which were only partially under Sauron's influence, the Ringwraiths and the destruction of Barad-dûr.
Firstly, with regards to the abilities of the Rings of Power. We learn a little about what happens to the Three after the destruction of the One, namely that they became powerless and were carried across the sea to Aman.
[After the desturction of the One Ring] it was made plain that the power of the Three Rings also was ended, and to the Firstborn the world grew old and grey.
The Silmarillion: Of the Rings of Power and the Third Age
When the Great Ring was unmade, and the Three were shorn of their power then Elrond grew weary at last and forsook Middle-earth ...
The Lord of the Rings, Appendix A, Annals of the Kings and Rulers: The Tale of Aragorn and Arwen
From the fact that the Three were only indirectly created by Sauron, given the ending of their power we can safely assume that the power of the 3 surviving Dwarven rings and the Nine had lost their power and, if they'd survived, become ornamental pieces of metal.
Secondly, as for the state of the Rings and the Mithril, assuming they'd both been kept in Barad-dûr¹, things are a bit more tricky. We know that the foundations of Barad-dûr were destroyed, and the tower completely broken and collapsed; we're also not told of people plundering the remains, assuming everything had been completely destroyed. The only real insight we get into the destruction comes from a rather vivid vision of Sam's:
A brief vision he had of swirling cloud, and in the midst of it towers and battlements, tall as hills, founded upon a mighty mountain-throne above immeasurable pits; great courts and dungeons, eyeless prisons sheer as cliffs, and gaping gates of steel and adamant: and then all passed. Towers fell and mountains slid; walls crumbled and melted, crashing down; vast spires of smoke and spouting steams went billowing up, up, until they toppled like an overwhelming wave, and its wild crest curled and came foaming down upon the land. And then at last over the miles between there came a rumble, rising to a deafening crash and roar; the earth shook, the plain heaved and cracked, and Orodruin reeled. Fire belched from its riven summit. The skies burst into thunder seared with lightning. Down like lashing whips fell a torrent of black rain. And into the heart of the storm, with a cry that pierced all other sounds, tearing the clouds asunder, the Nazgûl came, shooting like flaming bolts, as caught in the fiery ruin of hill and sky they crackled, withered, and went out.
Return of the King - Book VI, Chapter 3: Mount Doom
We get two interesting pieces of information here: we get a bit of information into just how immense the destruction of Barad-dûr was, as well as the withering of the Nazgûl and their end. Firstly, Barad-dûr is described as towers falling and mountains sliding, with walls melting and crumbling as well as massive clouds of smoke and heat. From the above we can, in my opinion, safely assume the total destruction of everyone and everything in Barad-dûr with fires hot enough to likely have melted all the Rings and Mithril that were left in the tower; however, it is also possible that they were merely buried in the wreckage. While someone could certainly have gone in and searched for the Mithril and rings afterwards, it is unlikely they would want to scavenge through the ruins of the Dark Tower.
Finally, the death of the Nazgûl is recounted in The People's of Middle-earth, in an earlier version of that which was published in the Lord of the Rings
After a lapse of 969 years Aragorn, son of Arathorn, 16th chieftain of the Dunedain of the North, and 41st heir of Elendil in the direct line through Isildur, being also in the direct line a descend ant of Firiel daughter of Ondohir [> Ondonir] of Gondor, claimed the crown of Gondor and of Arnor, after the defeat of Sauron, the destruction of Mordor, and the dissolution of the Ringwraiths...
History of Middle-earth - Volume XII, The People's of Middle-earth
This combined with the above shows that the Ringwraiths had been destroyed. Their dissolution would not necessarily mean that they were entirely destroyed: they were most likely left as houseless spirits waiting the calls of Mandos to go over the Sea and travel to wherever the spirits of Men go after death.
¹ The location of the Nine during the events of the Second and Third age is an interesting topic. One answer on site suggests that they were kept by the Ringwraiths themselves; however, this seems to contradict what Tolkien has said on the matter in both the Unfinished Tales as well as the Letters:
the Ringwraiths ... had no will but his own, being each utterly subservient to the Ring that had enslaved him, which Sauron held.
Unfinished Tales of Númenor and Middle-earth
While Sauron certainly kept the surviving three of the Seven, and it is likely that he kept the Nine, it is unsure whether they were all found in the same place.
After the end of the war of the rings, there was no use left for the other rings of power. As Sauron was died, his malice died, all those things that clinged might also have withered. The 4 rings procured by Sauron wold have been destroyed, of the nine rings of men they were destroyed with the Ringwraiths or Sauron himself.
The rings that had survived were now powerless.