Diagnosis
The magnetic charging contacts on the bottom of the camera or on the dock are dirty or corroded, breaking the connection.
How to Fix It
Power everything down. Inspect the four small metal pins on the bottom of the camera and the corresponding contacts on the dock. Use a cotton swab dipped in a tiny amount of isopropyl alcohol (rubbing alcohol) to gently scrub both sets of contacts. Let them dry completely for a minute before trying to dock the camera again.
If That Didn't Work
The battery itself might be completely dead and unable to take a charge. Try plugging the camera directly into the wall with the USB power cable, bypassing the dock entirely. If it powers on while plugged in, the dock or its cable is the issue. If it doesn't, the battery or internal charging circuit is likely gone.
Heads Up
A pulsing white light means it's trying to charge but can't. If you leave it in this state for days, you can permanently kill the battery. If direct USB power works, just use it that way as a stationary camera.