Why is quixotic pronounced as it is?

Since "quixotic" was coined with Don Quixote as its basis, why is it pronounced "kwicks-OTT-ick" when it should by rights/origin be pronounced "Key-HO-tick"?

It even sounds more onomatopoeiatic the latter way, as it resembles in its sonorous qualities "chaotic," which suits the subject.

Or is trying to make sense of English pronunciation a quixotic quest?

It appears to be dialect issue. Its not a word that comes up much in daily conversation, but the few people I know who use it pronounce it closer to your second (your supposedly "correct") way. More like KEY-hot-ick (emphasis on the first syllable, short o in the middle). However, there are comments below to the effect that your first pronounciation is reported as the "correct" one in the OED, and is understood as such in the UK.

The people I have heard use it are American Midlands dialect speakers (both Northern and Southern varieties). I suppose its possible that your dialect area (from your user info, I'm guessing California English?) tends to use the more normal Anglicized version.

Probably the reason for that first reported usage is that it is the pronunciation an English speaker would naively expect for that assemblage of letters, in the absence of any other information. The most common word starting with "qui" is "quick", which is pronounced with the same "kwi", and an "X" in English usually produces a "ks" sound. So if you didn't know the word derived from the name of a character in a Spanish novel, you'd expect it to be pronounced your first way, not in a Spanish-influenced way.

/'kwɪksət/ is clearly an anglicization of the Spanish spelling.
It's equally clear that such spelling pronunciations have always been very common.

This one, in particular, can be seen in action at the very end of
Canto 13, Stanza 10 of Byron's Don Juan
  (a title, incidentally, pronounced /dan'dʒuwən/ by the author, as the poem makes clear)

Redressing injury, revenging wrong,
  To aid the damsel and destroy the caitiff;
Opposing singly the united strong,
  From foreign yoke to free the helpless native: --
Alas! must noblest views, like an old song,
  Be for mere fancy's sport a theme creative,
A jest, a riddle, Fame through thick and thin sought!
And Socrates himself but Wisdom's Quixote?

Note the rhyme: wrong - strong - song alternating with caitiff - native - creative,
and ending in a couplet, with its rhyme boldfaced above. In this couplet,
Quixote has to be pronounced in two syllables, not three,
and the last two syllables rhyme with thin sought.

I.e, /'kwɪksɔt/. And from there to /'kwɪksət/ is no distance at all in modern English.