I don't want to put a dampener on your hopes, but I never lost any weight bfing. I thought the same thing as you when I started, but I did it for 6 mths (not exclusively bfing, but mostly bf) and didn't lose a kg without some effort on my part re: diet and exercise.
But you are only 5 wks into being a mum and bfing, so give it time. Try to relax and not worry about it, just enjoy your bub and think about it later.
As for your milk, thats probably adding to your weight too if you have so much.
HTH