Maybe you are bothered too much with the translations "know" and "understand". I feel that these translation are not that bad, but they may only be reflecting some shades of the meanings, and do not match precisely. The distinction is like this:
- 知る: To objectively come to know (have in mind) an established fact. (知っている is 'to know'.) Does not require thinking (computation); it is just about memory.
- 分かる: To subjectively grasp/accept the mechanism/reasoning/meaning behind something, or being able to handle/analyze/conclude something through thinking. As the kanji indicates, this verb is related to
分ける
, which means 'to divide', which leads to 'analyzing'.
In case of a question given, if it is something you need to think of or if it is a question that you have heard for the first time (which is the majority of cases), then 分かる
/分からない
is appropriate. If it is something that you either simply know or not, then 知っている
/知らない
is appropriate (Be aware that it is not 知る
/知らない
).