It can be maddening to get my daughter to eat sometimes. One day she'll love something, the next she won't touch it. When it's getting close to meal time I don't let her snack or drink anything at all so she's as hungry as possible. When it's time to eat I usually feed her what I think she needs the most first (usually veg) and will give her the rest after she's eaten what I think is a suitable amount. I'm lucky in the sense that she doesn't really like junk food, but she's so tiny (5 percentile height and weight) that I feel like I need her to eat as much as possible. What's helped for me is to sit there with her the whole time, kind of play with her and feed her/ let her feed me, and not treat her puny appetite like she's doing something wrong.
You're taking on a hell of a big project with this woman with all she has going on with her kid and her ex, and it sounds like you're handling it all well. But don't make the mistake of thinking anything will be a quick fix because you're looking from the outside in and feel like you can change something because common sense would say it's the right thing. Kids defy common sense and single parents have so much on their plate they may miss some things or lose sight of details because they have so many responsibilities. It's really hard being a single parent and you have your work cut out for you if you're really willing to step up and be a maing figure in the kid's life.