Other answers have covered the bases on quixotic quite well, I think. I just want to note here that quixotic isn't the only Spanish-derived word containing the letter string qui that (at least some) British English speakers pronounce in a very un-Spanish way.
In the United States, children are taught in elementary school about the Spanish conquistadors, who explored (and of course conquered) large areas of the New World during the very late 1400s and the 1500s—and the only pronunciation I have ever heard used here in the United States for the word conquistador is kunkeestuhdor, with the accent on the kees syllable.
But as this video of Procol Harum performing its song "Conquistador" in 1977 demonstrates, Gary Brooker, the band's (English) lead singer, pronounces the word kunkwisstuhdor, with the accent in the kwiss syllable. This, it seems to me, is entirely consistent with pronouncing quixotic as kwiksottik with the accent on the sott syllable.
Indeed, if we accept the three letters qui in Spanish as consistently receiving the pronunciation kwi (with a short i sound) in certain forms of British English, it seems to me that pronouncing the following x in quixotic as ks rather than as h follows as a matter of course, given the strangeness (to the mouth of an English speaker) of pronouncing the h sound immediately after a short i sound. In English we have several words that involve pronouncing an h immediately after a long e sound (beehive, knee-high, freehold, and hee-haw, for example), but none I can think of that involve a short i immediately followed by an h sound.
For that reason, I would expect an English speaker attempting to pronounce frijoles either to try to match the Spanish pronunciation with something like freeholeez (with the accent on the ho syllable) or to keep the i short and pronounce the word something like frijoleez (with the accent on the jo syllable) or frijolz (with the accent on jolz).