iPhone 5 Power Button Replacement Programme
Two facts that might disappoint you:


The damage done on your iPhone may be responsible for the broken power button. 
The iPhone may not have been opened before (by non authorised personal)


I just got my phone to the Apple store for a power button replacement, I had have a small crack in my screen, but never replaced it. They looked for water damage and told me I was able to let them replace my power button. They did not mind about the small crack.