1. It's better to avoid apostrophes in a formal letter, you should use the longer form, so instead of 'I'm' write 'I am', and instead of 'didn't', use 'did not'.
2. If you don't know the name of the person, end your letter with: Yours faithfully,
3. I am writing TO you in order to... (I think 'e-mail' is unnecessary)
4. We HAVE made A bank transfer...sending the documents requested
5. have not received a payment response yet.
I'm not sure that all of my advices are correct, since I'm not a native speaker. But I hope I could help you.
- For Teachers