Dear Gab, I hurt for you too. Been there, done that sort of thing. The pain is probably more for the fact that he didnt' show for your daughter's birthday. Anger is due to come up next.
This is where alanon would be a big help to you. We've tolerated so much crap from them over the years, we've got years of emotional turmoil inside. Let it go. Little by little. Detach. If he wants to argue, walk away...detach. If he doesn't show up, take your kiddies and go out and have a good time. Go to the playground. This will take your mind off him and his inability to be a human being and you can focus on your children having fun. Then top the day off with an ice cream cone. Always a winner. When my ah didn't come home as promised, it was a jump into the car to go to baskins robbins. Got to the point our son would tell me he hoped dad wouldn't come home so we could go get ice cream and ride the swings at the rec center. When I saw how happy my son was it helped me forget the pain and anger I felt because he once again screwed up and chose booze over us.
Be kind to yourself. Take baby steps.