It is supposed to be physiologically impossible to eat nothing and yet put on weight. Maybe you should give the forum more information. I cannot advise a T1 since I am a T2 but I still find your story odd in that you have gained weight.
I think you need to show a menu for what you are actually eating and drinking as there are so many interpretations of a low carb diet. This might help members with suggestions.
If you have been exercising then you have been toning your muscles and muscles weigh heavier than fat. As your friend has not been exercising that could be the reason she has lost 12lbs as she has not toned up her muscles. Have you noticed any improvements in the shape of your body?