Similar to "Nan nen?", "Nan nichi?" can mean both "How many days?" and "Which day?". See the following example.
A: 来週お休みをいただきたいんですが。
Raishū oyasumi o itadakitai n desu ga.
B: 何日?
Nan nichi?
In this case, B's reply "Nan nichi?" is ambiguous. It can mean both "How many days do you need?" and "Which day do you want to take off?", thus, it's hard to say that "Nan nichi?" is the best chice for "How many days?" here. Instead, "Itsu?" works better as "Which day?" than "Nan nichi?", and similarly, "Dore kurai?" can be better as "How many days?" than "Nan nichi?". Furthermore, there are many variations for "Nan nichi?" in the level of politeness. The most appropriate expression differs depending on the relation between the speaker and the addressee, for instance, there are cases where the polite form "Nan nichi desu ka?" should be used. This is exactly why the context is needed for the correct answer, as Julie-san pointed out.