I think you're doing great! I'll dissect your translations so that you may learn, but please don't be disheartened :-) I'm not an expert myself, but growing up hearing Korean through parents, relatives, and TV dramas, I have a sense for things that definitely don't sound Korean.
Let's start with
세상의 깨지기는 느낍니까?
You correctly begin with "의" for a possessive form of "world".
The "기는" here is not quite right. You would normally only see that form when trying to incorporate the complex word, "even," like
which means, "Did it even break?"
Broken-ness is a strange word, though. An adjective converted into a noun. In Korean, I recognize that as replacing a final ㄴ with an ㅁ:
- 아름다운 (beautiful) vs. 아름다움 (beauty)
- 강한 (strong) vs. 강함 (strongness)
There are exceptions of course, like 행복한, whose noun form is just 행복 (though 행복함 doesn't necessarily sound wrong). Google Translate gives me this:
- 부서진 (broken) vs. 부서짐 (brokenness)
But this word doesn't fit your lyrics, as you may have already thought. "부서진" describes something more physical and concrete, certainly never refers to emotions, and as for a world, invokes more a picture of post-World War III destruction, rather than society or relationships.
Your choice of 깨지다 is much more appropriate! This word is used for emotions, society, and relationships, as much as broken glass. The problem? I don't know if the following...
- 깨진 (broken) vs. 깨짐 (brokenness)
...actually works. The first is definitely a real and common word. But I have never heard the ㅁ-replacement on this word. It's probably "technically" a word but too strange-sounding for lyrical or even conversational use. For learning's sake, if it were a usable word, then your last phrase is good as-is!
세상의 깨짐을 느낍니까?
Anyway, maybe someone can point you to the correct word for "brokenness," but I wouldn't hold my breath.
I would try a different construction, which is actually the second path you were already trying:
세상이 깨지고 느껴요?
First, you seem to have switched to a more casual "느껴요" here. Whether to use "느낍니까", "느껴요", or "느껴" is a whole different topic, but let me revert to "느낍니까" to keep consistent with the previous tone. So,
세상이 깨지고 느낍니까?
Here, the piece that's off is just the middle again :-) What you've written roughly translates to,
The world breaks, and... do you feel?
The "고" by itself is a temporal "and," e.g. a "then." What you saw in your analogue was actually "-하다고", whose "고" is totally unrelated, and is some kind of a grammatical thing I don't know how to explain. The direct analogue of that to the word for "broken" is
So your combined sentence would be:
세상이 깨졌다고 느낍니까?
This is getting very close to your original sentence!
I think you're fine using the above, but if you need your translation to be a cut above the rest, I would say that 깨졌다고 is still a bit strong of a word. It's like... something definite has shattered. Whereas your lyrics, to me at least, evoke more of a feeling that something is wrong, but the mood lies in the vagueness of what's wrong. I'm sure an actual, daily user of Korean could come up with better words, but this is my best attempt:
세상이 잘못됐다고 느낍니까?
It doesn't sound very lyrical to me, but it translates literally to:
Do you feel the world has gone wrong?