Klintur is appalled by the sheer number of spiders and believes that killing off their queen may help reduce the damage they inflict upon the halls of Khazad-dûm.
'You may have seen the spider-queen who calls the nearby lair home. You can find her east of here -- don't hurt your legs jumping off!
'Defeat her, and my breath will come a little easier.'
The spider-lair is due east of the Shadowed Refuge, which is on the western edge of the Foundations of Stone.
Klintur has asked you to defeat the spider-queen who prowls the nearby spider lair.
Klintur is at the Shadowed Refuge, above the spider-lair.
You should return to Klintur and reassure him that Uanuiloth has been defeated.