It might be more accurate to say that Newton placed a theoretical foundation under Kepler's solution. (Though Kepler thought planets orbited the sun, not the center of mass of the sun-planet system. Thanks for clearing that up, Isaac.)
It is also inaccurate to say that Newton did not solve the three (or many body) problem. The answer today is the same as it was in Newton's day, as my Gujarati Calculus professor would say, "Integrate, only." It's like Woody said about Buzz Lightyear, "That's not flying, that's falling with style." Same thing, these are not closed from solutions, they're integrating with style.
All the hubbub here is about 13 new solutions. I wonder if anyone has put an upper limit on how many there are?