Sump pumps don’t turn on when they are full, that is the most common problem with them. This is caused by a problem with the float inside the sump pump. The float rises as the sump pump fills with water and triggers the pump switch when the pump is full. When the pump empties, the float falls again. But if the float is damaged, water can leak into the float, keeping it from rising with the water level.
Inspect the rod attached to the motor on the top of the sump pump. Move it up and down manually to see if it moves smoothly. Corrosion will keep the rod from moving smoothly. Clean the float road with steel wool.
Inspect your sump pump's check valve. It has an arrow on it. The arrow should be pointing away from the pump. If it's pointing towards the pump, remove and reinstall the check valve.