Diagnosis
The tactile membrane switch behind the power button is likely failing or has become unresponsive due to wear or a minor manufacturing defect. It's a known weak point on this model's joystick control assembly.
How to Fix It
Unfortunately, the power/joystick nub is a single unit. Before you go deeper, try pressing it more deliberately—use the tip of your finger and press straight in, not at an angle. If it's consistently unresponsive, your real fix is to use the monitor's auto-wake feature. As long as the PC is sending a signal, the monitor should turn on automatically when you wake your computer. To turn it off, you can just shut down your PC and the monitor will eventually go into standby.
If That Didn't Work
If it's under warranty, contact LG. If not, and it's truly unusable, the control board (which the joystick is part of) would need replacement. It's not a user-serviceable part without disassembling the monitor, which voids warranty and is tricky.
Heads Up
Don't keep mashing it harder—you could damage the internal connection further. The auto-wake/sleep with your PC's power state is the best workaround.