So patch 6 changed this spell being able to put traditionally sleep immune characters like elves asleep. Not gonna lie though I sort of loved that this spell was an exception to that. First, because it's a sixth lvl spell. Sleeping the sleep immune felt like a worthwhile cost of the spell slot. Second, the fey ancestry perk reads to me like it is geared more toward enhancement and illusion charms common to the Feywild; however Eyebite is a necromancy spell. I personally see it more as simulating a "sleep of death."

If there are any readers of The Fellowship of the Ring, In the chapter Fog on the Barrow-Downs, the hobbits are abducted by Barrow-weights: "It leaned over him. He thought there were two eyes, very cold though lit with a pale light that seemed to come from some remote distance. Then a grip stronger and colder than iron seized him. The icy touch froze his bones, and he remembered no more."

This is pretty much how I picture eyebite sleep happening.

Unrelated fun fact, in this chapter of the book, the barrow-weight uses an incantation (for the death sleep) that goes:

Cold be hand and heart and bone,
and cold be sleep under stone:
never more to wake on stony bed,
never, till the Sun fails and the Moon is dead.
In the black wind the stars shall die,
and still on gold here let them lie,
till the dark lord lifts up his hand
over dead sea and withered land.

Fans of the movie might recognize some of these lines coming from Gollum in The Two Towers.