Even if your friend was "just" fat you should never have said something so nasty to her. That's not how friends behave! And it wasn't just a one off sarcastic comment, by your own admission you have made fun of her for quite a while.
My friends and I do take the mick out of each other, but never about appearance, that's just a step too far.
I think you have to admit you were wrong to say these things to her and ask her to forgive you. If she says no, then you will just have to suck it up and learn not to be so mean to others in future.