After hearing it several times I discerned the meaning as something along the lines of, ‘How should I know?’. I can think of numerous ways to say this in English with varying strength and feeling – e.g. ‘How the hell should I know’ – but I’m not sure where 知るか goes on this scale. Perhaps anywhere along the scale, depending on the emotion you put into it.
Today, for the first time outside the realm of anime, I heard the phrase used in real life. A student was asking a question to another student, who calmly and coldy replied… 「知るか」.