Concept:When the verb "like" is used to say someone enjoys or does not enjoy an activity, the next verb takes the \textit{-ing} form (gerund).
Explanation:The given sentence is a general statement: she does not enjoy the activity of writing letters.
The verb to change is "write".
Its gerund form is "writing".
So, the gerund phrase "writing letters" becomes the object of "like".
The base verb "write" cannot be used directly after "like".
The continuous infinitive "to be writing" does not fit naturally in this simple present-tense sentence.
In such patterns, verbs of liking and disliking are commonly followed by the gerund form.
Thus, the correct completion is: "She does not like writing letters."
Answer:D. writing