Game Center has stopped backing up my games ... What is the fix?

[iMore Question]

Every time I opened a game app, I would be greeted by my Game Center name and the games, I assumed, were being backed up in Game Center. Two months ago, I was no longer being greeted and I suspect the games are not being backed up. My iPad is broken (it refuses to charge up) right now and is scheduled to be replaced in a couple days but I want all of my games backed up on to my replacement device.

 

[phreddyl]

If it refuses to charge up I'm not sure there is much you can do. Maybe it backed up anyway.