"Writing a letter is fun."
Translation:편지를 쓰는 것은 즐거워요.
Yes it should come first. That whole phrasing goes together 편지를 쓰는 것은 because that's the subject. Writing. 쓰는 것은. Writing a book. 책을 쓰는 것은.
The gerund has to follow the same subject-object-verb order as it would in a full sentence. If you said "I write a letter" then the verb to write would come last.
This sentences technically translates to "(i) will give you the letter (i) wrote". 즐거워요 means "will give". I feel this is wrong.