I've heard writing "thank you" increases the changes of getting a gift. I always write "おりがとうございます!" but sometimes I only hear back from one or two villagers and they don't always give me presents. I always mail them a present too which helps; usually fruit or unwanted shirts and furniture.
For the letter badge, your letters must be at least 10 characters long.
Edit: I looked at that guide and it says for a short letter (one line long), you get fruit and clothes, but my one line letters in New Leaf (that guide is for WW) have gotten me furniture. It's still a helpful guide.
Remember to use the small circle that's the period mark in Japanese. I write letters whenever I can. A little bit of English is okay as the villagers use some too.
I usually write このふく きてください. Which means "please wear this shirt."