Why does the hot cooktop indicator light stay on after the cooktop is cool? The indicator light now wont turn off

This indicator may be driven by some form of temperature sensor. If the light is staying on too long; or indefinitely, there is likely something wrong with the sensor, the sensor wiring, or the control board that interprets the signal from the sensor.

Without knowing the make and model of the stove, it's impossible to tell you where the sensor is, what type it is, how exactly to troubleshoot it, how to replace it, or if it can be adjusted.

The basic idea is that when the stove top heats up, the temperature sensor sends a signal either directly to the light, or to a control board. This causes the light to come on, and stay on as long as the stove top is above a certain temperature. When the stove top cools down, the signal is no longer sent, and the light goes out.

You may want to get an Appliance Repairer, they are worth it. Otherwise, you need to take stuff apart (with the breaker[s] off) just to know the right part to get. These are symptoms of a bad switch, it could be the $20 switch or the $90 switch... see