If you haven’t replaced the battery that is where I would start as its likely long worn-out even if you leave it plugged in all of the time.
The front white LED is sleep indicator light (SIL) here’s a bit more on it Mac computer Sleep Indicator Light behavior.
The system will also go into sleep when its not getting enough power from either the charger or battery. If the MagSafe plug is Green and flashes five times when starting up or after waking from sleep then the battery is in need of replacement for sure. Amber implies the battery is in need a charge and if it never gets to green then that too implies the need for a new battery as well.