Will Apple replace my Iphone 5 broken sleep button?

It's just a little over a year old and is not damaged in any other way. The top button use to sometimes work and then only in the left corner but now it's completely dead.

Possibly. Call them and ask but for now use the assitive touch in the settings options

If it's still under warranty then sure why not. If you can't do that, then try having it fixed by a technician. If not, you can sell it and get the iPhone 5S. You can check out eCycle Best. They'll take in broken iPhones.