The Lidless Eye refers to the eye Frodo and Sam see when they climb Mount Doom:
Far off the shadows of Sauron hung; but torn by some gust of wind out
of the world, or else moved by some great disquiet within, the
mantling clouds swirled, and for a moment drew aside; and then he saw,
rising black, blacker and darker than the vast shades amid which it
stood, the cruel pinnacles and iron crown of the topmost tower of
Barad-dûr. One moment only it stared out, but as from some great
window immeasurably high there stabbed northward a flame of red, the
flicker of a piercing Eye; and then the shadows were furled again and
the terrible vision was removed.
That you are "left naked to the Lidless Eye" in the Houses of Lamentation suggests they are a dungeon in Barad Dur. The tone suggests a waiting place before or after death, a sort of purgatory. There are multiple places like this, like the Halls of Mandos, where Elves wait until they reincarnate. Tolkien Gateway also mentions the last words of Thorin:
'Farewell, good thief,' he said. 'I go now to the halls of waiting to
sit beside my fathers, until the world is renewed.'
The Witch King is invoking images of eternal dread to frighten his listeners.