(con) If you still want to try it then, you could try offering the food, or you could make your own baby food with a blender & add kale to that. If you google, there are lots of articles & such on how to make your own baby food as it's a pretty popular thing to do, especially recently with the whole natural parenting movement. As just a supplement, the corn in the Beechnut baby food likely wouldn't be a huge issue, especially because of the consistency (takes away issues with choking or chewing, also likely makes it easier to digest), but I would still avoid it right now due to the kale & spinach.
I would skip spinach, especially right now. It's very high in oxalates, which a) bind with calcium and make in unavailable (so that kind of puts a dent in the argument that spinach is high in calcium) and b) oxalates are thought to contribute to kidney issues.
I know you want to get other foods into her, but I would focus on a balanced wet food right now rather than a lot of baby food, especially if you're going to have to syringe her anyway. If you want to try offering a bit to see if she'll eat on her own, to supplement, that's fine. But focus on the wet food so she's getting something already balanced along with being higher in moisture.
Kale is moderately high in oxalates as well, so I would still hold off until you have her current issues under control and figure out what's going on.