Discord takes place in any relationship even seasoned ones. The fact that after 5 months you had started fighting is not unusual (people get used to each other, tension may build up, pleasing and impressing the other person is not longer a priority so, it happens) What also can happen though, if the two people are mutually invest in the relationship they both look for ways to improve it.
It seems that this did not take place on his part which can be seen as either immature and or that he truly is ready to leave the relationship.
On your part, you still find hope and hence want to get back with him. That may not be healthy. Of he wanted to come back what would stop him? It is a good idea to write the letter and express how you feel. Bottling feelings up is never healthy. As far as change, if he wants to work on the relationship, both can consider couple's counselor. Change is a very hard thing to do especially on your own. Also, you may have to see if he is willing to hang around while you are working on your change. Try to keep your expectations realistic and know that you like any one else is worth of being loved and being happy. When a man wants a woman he does not let her go.