Kepler did make progress - he showed that equal area covered by the planet on the orbit implies that the orbits are elliptical (and vice versa). Enough progress for the next guy to take it further.

The next guy was Newton - he knew the solutions to a central force problem (thanks to his invention of Calculus - notation for calculating with infinitesimals). Movement with conserved angular momentuum <=> movement with the equal-area property <=> happens for a force ~1/r^2. From that he concluded that the central force is "gravity". The deriveration is simple enough for 1st semester physics student, but you will need to cleverly change variables once or twice to get a familiar integral.

https://en.wikipedia.org/wiki/Central_force

https://en.wikipedia.org/wiki/Classical_central-force_proble...