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
分からない is appropriate. If it is something that you either simply know or not, then
知らない is appropriate (Be aware that it is not