I agree with James and Michael.

The name of the book/character is pronounced 'Kee-hoe-tay' these days, making the whole name sound something like 'Donkey Hotay'.

The thoroughly anglicised pronunciation is only used for the adjective -  pronounces 'kwiksotic', to rhyme with 'exotic'.

Now, here's a question. How do you pronounce 'Quinoa'?

I've always said "Don Kee-ho-tay", but  I just asked my husband, who's never taken Spanish, and he said "Don Kee-ho-tee." I'm not sure I've ever said "quixotic" out loud, but if I did, it would be anglicized.

I pronounce "quinoa" as "keen-wa".

That's a very good question, Su.Ki. My wife who "discovered" the stuff (which is being heavily promoted in the U.S.) thinks it is pronounced "Keen-wa." However, a Mexican language partner did not recognize it when I pronounced it that way, and asked if I mean "Keen-oh-a." However that was a side issue and I didn't stop to be sure--it was actually a discussion of whether I was preparing a hot drink called atole, made with oatmeal and amaranth, correctly. 

Ahdictionary.org is showing both options: (kĭ-nōə, kēnwä). 

Where is the stuff from? Click:  "Andean region of Peru, Bolivia, Ecuador and Colombia", https://en.wikipedia.org/wiki/Quinoa , and it's Quechua word. So perhaps people in Mexico, who are tuned into Nahuatl but not Quechua, use a "hispanicized" pronunciation or something. 

I would expect to hear kee-hoa-tay and kwik-so-tik as well in the UK, so it's almost the same as in the US.

However, they are not common words.  I can't imagine hearing : "Our Wayne's been behaving a bit quixotically recently - I just don't know what to do with him!"

In my experience, people who are familiar with the words are likely to have some idea of pronouncing them correctly.