How can I calculate the time between sunset and actual darkness? I would like to know the time it takes to go from sunset to dark based on GPS location.

In addition, ideally I would also like to have a formula that determines the percentage of sunlight at each minute past sunset. For example, after 15 minutes, there is only 50% sunlight left.
One may define three forms of twilight on Earth.  Although the actual amount of light depends on weather, topography, and land cover, they are defined as:


Civil twilight: Solar angle > -6°
Nautical twilight: Solar angle > -12°
Astronomical twilight: Solar angle > -18°


Human eyes see logarithmically so % of sunlight left is not a very useful measure, unless you are interested in incoming shortwave radiation for solar panel or other energy calculations.

To answer your question, you need to calculate at what time the Sun will be at those elevations below the horizon; this is calculated in a similar fashion to sunrise and sunset times.  However, on top of that you also need to consider atmospheric refraction.  See position of the Sun and sunrise equation.  The formulae are quite complicated, but you may be able to simplify them if you're only interested in the duration, not the absolute times.







The time after sunset at which those occur are a function of latitude and time of year.  On this carpet plot from Wikipedia you can read the length of different forms of twilight at 70°N (north of the Arctic circle):