The correct option is B How long / to wash
The context refers to washing dishes. Here, the query refers to the time taken. So the first fill up will be "how long" as the literal meaning of 'long' is length or duration. Thus, the duration of time it took for an activity is indicated by its use.
The "to" infinitive is used with the verb to indicate purpose. So, "to wash" will be the correct answer.
Therefore combining the two correct options, the correct answer is option B.
"How much" refers to price of something.
"Washing" is the noun form which names the activity. Here, the infinitive form of verb is required.
Thus, options A, C and D are incorrect.