I'm always conflicted on plural/singular translations on this course. On the one hand, there's no 들 particle so there is no explicit plural implied here. On the other hand, it's quite unnatural to talk about general activities in the singular form (Writing letters is fun vs Writing a letter is fun). Is this a difference I'm not used to between 쓰기 and 쓰는 것? Or should both answers be acceptable?
No the difference has nothing to do with that. Mainly because there is no difference in Korean, and the main English sentence should be as you suspect "Writing letters is fun."
-기 and -는 것 are basically the same but are habitually used differently only in certain circumstances. In this case, talking about doing some action, it is more common to use -는 것. This Korean sentence is a good example of it. Another one could be 저한테 피아노를 치는 거 어려워요. = Playing piano is difficult for me.
Adding -기 is the same as making it a gerund but it does feel wrong to say 편지를 쓰기는 즐거워요. I do see it used a lot in titles, like "Chapter 1: Starting a Business" = 사업를 시작하기. It's also paired with certain verbs, like 바라다. 너 도와주기를 바라. = I wish you'd help me. 성공하기를 바랍니다 = I hope you succeed.
Maybe there is a rule with more specificity but in the meantime, try to get a feel for what sounds right. You'll hear it if you watch Korean films or sample dialogues or something.