Explain to her that her weight is normal. The main thing is not to say "I like you like that", it sounds offensive. That she has no problems with weight. That being skinny is ugly and unhealthy.
I think that persuading is not the best way. Take part in her self-improvement, without fanaticism and pressure. Surely, she wants to change herself because she feels that she wants to change. This is her choice and her body. Your help is not to interfere with her in this.
Look for stories about ex-anorexic women, if she has not yet entered the state of "I don't care, I have to lose weight at any cost", it might make her think. The mortality rate from anorexia is the highest among mental disorders. Those who survive often have major health problems, often incurable diseases, and are on pills. Often then they cannot have children at all. And anorexic women are either former or dead, this is not the state in which a person lives a happy life. Look for information on diets not working. Logically, they should not work, because this is only a temporary measure. This can help you lose weight quickly, but to stay slim you need to change your regular diet, otherwise everything will return. Maybe it will help to learn how to cook healthy dishes from the menu of proper nutrition, and treat them to a girl, persuading that there is definitely no harm from this. Pamper sometimes with something tasty. Compliment more often.
Nothing. Now most girls sit on TA (or similar publics) and they are hammered into their heads that being thin is extremely beautiful, and if you don't weigh 35kg, then you are fat. I do not argue, yes, there are beautiful and thin girls, but there are also complete beauties. And those and those ideals of female beauty. There is no limit to perfection. Better, guys, drive your loved ones that you need to read books, think with your own head and listen to you.