Translation:The book is not good but I like it anyway.
Basically because you have introduced the en word "boken." Without boken you would say:
"Det är inte bra men jag tycker om det ändå."
Essentially, once you introduce a word the later uses of "it" take on the gender of the word you're talking about (I admit though, I don't know how far that extends though. I don't know if you would revert to det several sentences later, but within the sentence it takes on the word's gender).
inte X men Y means there's an opposition of the type 'not X but (instead) Y'. Another way of saying it in English is 'but rather'.
Jag dricker inte te utan kaffe I don't drink tea, but coffee. (I drink coffee sort of 'instead of' tea)
The second thing sort of 'replaces' the first thing.
But 'liking the book' is not a replacement of 'thinking it is good' in this sentence.
You could say this:
Jag tycker inte boken är bra, utan jag gillar den
but that would mean: I didn't mean that I think the book is good, I mean that I like it.
And then there wouldn't be an ändå in the sentence, because there'd be no place for 'anyway' in that sentence. – Hope this helps!
I wonder if Duolingo makes known how they do this or whether there is an industry standard practice among language-learning platforms. Although I am only learning the language, to me the rhythms and inflections of Swedish phrases and sentences in these lessons sound convincingly human -- in such fine details that are beyond the capability of rule-based text-to-speech synthesis, I think.
Although all timing and inflection information can conceivably be entered manually for each individual phrase and sentence, it would be easier for someone just to speak into a microphone instead. I surmise Duolingo then processes these recordings so that all phrases are heard in a unified signature voice, even if the original recordings were of disparate speakers. So, even though the "voice" has a computer-generated quality, the rhythms and inflections must still come from some native speakers behind the scene.
Nevertheless, someone still might have made a mistake. So, it is good that Will asked and Lauri checked.
"Anyway" usually has the same meaning as "despite that" ("trots det"), and "ändå" is a good translation for that, in both directions. An alternative is the set expression "i alla fall" (lit. meaning "in every case") that can mean "anyway", but can also mean "at least". "Ändå" can also be useda differently, getting a meaning similar to "since"
"Yet" in your example, and "still" could also be used like that, also means "despite that" in that sentence, so it would be "ändå" here. But the more common use of "yet/still", meaning "up until this point in time", in a negative/positive context (e.g. "he is not here yet" / "he is still here"), is better translated to "ännu" / "fortfarande".
Is "ändå" like "nevertheless" and "fortfarande" more like "continuing to be"? e.g., Is "He is still sleeping" translated as "Han sover fortfarande" rather than Han sover ändå"?