DIAGNOSIS: A faulty electrical connection between the charger base and the handle, despite power reaching the base. The most common point of failure is the contact points.
FIX: Unplug the charger. Thoroughly clean the two raised metal pins on the charger base and the two corresponding metal contacts on the bottom of the handle with rubbing alcohol and a cotton swab. Scrub off any green/black corrosion. Ensure the handle is seated directly over the pins, not at an angle.
IF THAT DIDN'T WORK: Try the charger in a different outlet. If the base light stays on but the handle still won't charge, the charging circuit inside the handle is likely faulty. The base itself rarely fails; the problem is usually in the handle's contacts or battery.
HEADS UP: Don't use the brush immediately after placing it on the charger. A completely dead battery needs 30+ minutes of undisturbed charging before it can even power the light.