By noting mathematical relations in the results of the sky-staring, and developing a theoretical model that explains them.
Kepler stared at the sky a lot and using little more than complicated geometry and trigonometry discovered his
laws of planetary motion. The orbits are ellipses. The orbits sweep out equal areas in equal times. The period of the orbit is related to the size of the ellipse. These are observational regularities. Why do these regularities exist? Kepler didn't know. But it was a work of genius to even discover them through careful observation.
The genius of Newton was to see that a theory of gravity with an inverse square law
explains all three of Kepler's Laws.
You don't actually know what you're talking about here. The fact that the name of the situation has 'problem' in it does not mean that the theory 'has a problem'. We can write the equations of motion for the general
two body problem in a closed form. We can't for the general three body problem. (Or n body problem). This has nothing to do with whether the theory is valid.