Hellospoon, I think the advice to simplify it is best. You need to state what it is you need them to do: e.g. bring the dog in by 8 so it is not barking outside, or make the dog bark less as the noise is excessive.
Your letter doesn't make it clear what you need them to do, nor what will happen if they don't.
I would say I hope we can resolve the matter between us, but if not, I will be reporting the dog barking to the local council noise officer.
The council do take this seriously, my landlady years ago was sent a letter from the council saying her dog barked excessively and that she had to stop it, he barked when no-one was home, so she ended up having to get sitters for him. His bark was terrible, though, it was just endless, so I am sympathetic.
I know this sounds silly, though, but why is your child being kept awake at night by the dog barking. Of course it is possible, I just wonder if there's anything you can do about that in the short term, move their room, let them fall asleep in your room, play a low music in the room etc. just to relieve the stress whilst trying to sort it out. Of course, it is not your fault the dog is barking, but it's unlikely they will be able to prevent it doing it completely, even if it is reduced/they bring the dog in at night.