You do have a reason - YOU.
Your child does depend on you for love, support, encouragement, but you have got to choose to end your eating disorder and lose weight only for yourself. It's not selfish at all to think like this, either. If you take care of yourself physically and mentally, you'll be the best you, the best mom, the best worker, etc etc that you can be.
Good luck in your journey, it's often overwhelming to look at it from the outside, but once you're on the path and you start making progress, it'll start to get easier. I think the best advice I've ever taken from this place is a quote (I don't know who said it though), "You don't need motivation, you need commitment. Commitment is doing what you set out to do, even when it's the last thing you want to do." I was always looking for motivation to eat right, not binge, and get healthier... But really, I needed the commitment. I slip up and make mistakes, but that's part of the process. You forgive yourself for making the mistake, you analyze why you made a mistake, make necessary changes, and keep moving forward. I think these things are absolutely critical to long-term ED control and weight loss.