First of all, I want to congratulate you for knowing that you need to lose weight and taking the action you need to do it. A 9 pound weight loss is a 9 pound weight loss and that is great. As the mother of 3 daughters who are also my best friends, I know how much you value your daughter being in your life. It is a blessing to have a daughter who wants to spend time with you.
HOWEVER, you are still the mom and she cannot disrespect you, in private or in public. I would venture to say that she feels threatened by your desire to get healthy and if, as you say, she is overweight as well, she probably feels a little pressure to do something about her own weight issues. That said, she has no right to sabotage your efforts or to put pressure on you to overeat when you are with her.
I suggest you sit her down and tell her that if you do not do something about your health right now you aren't going to be around in 10 years. Plain and simple! Tell her this is a matter of life and death for you. Maybe even take her to the doctor with you so she understands the gravity of the situation. Ask her to join you on this journey. Get her a bike and have her go riding with you. Take walks with each other....they are a great opportunity to talk! Take a healthy cooking class together so she can learn that "healthy" food can also taste great! Find new, healthy activities to do together.
She doesn't want to lose you and even if she doesn't realize it now, she will thank you for losing weight and getting healthy so that you can be around for a long, long time. Be an example to your daughter that it is NEVER TOO LATE to make a new start.
In the end, you may have to limit the time you spend with her if she can't support you right now. Don't go out to eat with her but rather focus on doing other things. Once she sees that you are serious and aren't going to be bullied into quitting, she will hopefully come around.
Good luck and