I'm no expert on coaster brake hubs. For instance, I can't name the various parts inside there. But I've rebuilt mine a handful of times, so I've learned something anyway.
I can't tell you what's wrong. But obviously there's something. I'd take it apart and look for damage to the parts that push and pull together to apply those brake shoes. There's on odd little part with a spring and some 'pawls' that fit into slots. I wouldn't be surprised if that's been damaged.
If you haven't got a source for used parts, someone in the neighborhood who keeps a 'bike junkyard' for instance, then I'd start looking for one. You're likely to need parts for that hub. If not now, then soon.
If you don't have a set of rim brakes on your bike, then I'd get one soon. That way you can share the braking load between the hub and the rims.
Best of luck with it.