Power button won't click

There's another component: [The Power/Sleep button bracket}(iPhone 5s Power Button Replacement). One possible answer is "no, that's not possible; the button flex cable doesn't make it 'click.'"

There's also damage in that area (the back glass is broken), so I guess that could be involved, but I don't see how.

@calion42 Well, the actual "click" action is caused by the metal dome on the circuit board. The external button compresses the dome causing it to deform and once it gets past a certain point it snaps down to hit the electrical contact pad on the circuit board; that's the source of the tactile and audible click you hear and feel.

So you do need all three parts in order for it to work; the physical button to depress the dome, the electromechanical dome/circuit itself and as you pointed out, the button bracket that supports the circuit flex. Outside of physical damage, there's not much to go wrong with the button itself and the bracket, but the button flex cable is subject to wear and will eventually quit working at some point.

@dadibrokeit That helps, thanks.