A: What . . . . . . . .?
B: She is writing a letter.

Statement 'A' is a question, so the sentence for the blank must be in an interrogative form. Statement 'B', the answer to the question in 'A', is in the present continuous tense and so the question must also be present continuous. 'B' tells us the current action of the subject 'she'. Thus, the sentence to be used in 'A' must be a question that asks about the current activity of the subject 'she'.
Option A is incorrect. Although the sentence is an interrogative form and also asks about the activity of the subject, it is not in present continuous tense. In other words, it does not ask about the current activity of the subject.
Option B is incorrect. It is an incomplete phrase as it lacks a main verb and also it is not in present continuous tense.
Option C is correct. It is an interrogative phrase in present continuous tense. Also, it is asking about the current activity of the subject. Thus, it meets all the required conditions.
Option D is incorrect. Although 'She is doing' is in present continuous tense, it is not an interrogative phrase, so, cannot be used to ask a question.