Get that power button fixed! There could be awkward workarounds such as using Project My Display in Windows Phone 8.1, but otherwise, look into repair You shouldn't try to live with such a device anyway. Repair will make your life easier.
There isn't a way to natively take a screenshot without the power button. The OS is designed with the assumption that your phone is functional. Supporting and implementing backup methods for issues like this wouldn't be feasible. Get your phone repaired/replaced.