You may wonder what the secret to playing beautiful solo piano ballads is. OK, I'll tell you, first there are three basic elements upon which all great music is built upon: Melody, Harmony and Rhythm. (The classical virtuoso Leonid Habro argued dynamics as a fourth and I agree.) So, as long as you can relax and play simple/wonderful melodies, solid harmonies and concentrate and breathe at the same time, amazing things can happen.
Now, there are many different approaches to Solo Piano, and very few Jazz pianists are masters of them all. One rare exception might be the great Dick Hyman, who can pretty much imitate the style of just about any artist, and do it better than the originator his or herself.
In terms of my own teaching focus, I try to get each and every one of my students to develop their own unique sound and niche. The only major steering I do is when I get a student who is clearly "immersed" in the exact same repertoire and style that every single other young musician is playing. At that point, I may play the role of contrarian and see what I can do to help the student find his or her own way of embracing some form of unique identity. (You might call me a copycat crusher.)
In the end, music is a collaborative journey. In our lessons, I'm not going to try to get the student sounding like me, or anyone else, rather we're going to explore options and set a path. By the end of the journey (yes, no journey is infinite), my hope is that the student will be way stronger on the 4 elements, and be well on the road of professing his or her core sound.