Kúli wants Nykr's death to be reported and celebrated in the true dwarf-fashion.
'${PLAYER}, Nykr's passing is a cause for great grief. I can think of only one way to properly celebrate his life, and death, in correct accordance with our most ancient and venerable dwarf-traditions.
'Nykr was friends with many on this expedition, and they are stationed from the Hollin Gate to the ruins of the Bridge of Khazad-dûm. Will you honour Nykr's memory with them, by quaffing a good ale with each of them where they are staying at the ten major encampments of my people throughout the Mines?'
You should visit, in no particular order: the Shadowed Refuge, Jazârgund, the Deep Descent, Anazârmekhem, the Dolven-view, the Chamber of the Cross-roads, the Orc-watch, the Twenty-first Hall, Durin's Threshold, and the Rotting Cellar.
Kúli wants you to speak with Nykr Locket-bearer's friends in each of the dwarf-outposts throughout Moria, telling them of his death and hoisting a memorial tankard in his name with them each.
Kúli can be found north of the Chittering Hole in the Redhorn Lodes.
Having muddled your wits with Nykr Locket-bearer's friends all over Moria, you ought to return to Kúli, his kinsman.