What would you do with a hamster who wont stop chewing bars? (read details)?

I have a golden Syrian hamster, with a multilayer cage. A few days after we got her last summer, we noticed that she was chewing the bars of her cage, and since have tried everything to dissuade her; we have given her chew sticks, apple wood hamster log type things, cardboard, forage balls... She wont chew the logs after she has taken the bark off, on the most part, and will easily get through a whole cereal box of cardboard in an evening, but does nothing with it, doesn't move it to her bed or anything. I'm now thinking that we have no option but to move her into a cage without bars, but I really dont want to do that, as she loves climbing. Whenever she comes out of her cage, she will climb anything we let her- me, cushions, obviously nothing she could fall off and hurt herself. And despite having a wheel (she does fit it, btw) she barely uses it, opting instead to climb and do acrobatics on the bars and levels of the cage. In addition, the only none bar cages I can find are much smaller. So what would you do to stop her hurting herself? Thanks!