Our ceiling is about 15 feet tall. We'd like to place LED lights on it to light our seating area and room. We're wondering if the lights are too far if the light may be too diffuse by the time it reaches ground level. Is there a formula, chart, or rule of thumb for what kind of lights you can put at different heights?
There is no general formula besides some really scientific ones. It is rather difficult to find rules of thumbs for calculating light in rooms. In your case the illumination levels on your seating area depend strongly on the beam angle of your luminaires.
However, it is both possible to do some calculations by using pen and paper and to do exhaustive simulations using a computer.
While it takes some time to get into it, you can yield very professional results. Both softwares contain a 3D-modeler for your rooms and exhaustive luminaire databases as well as the possibility to import lamp date like IES and EULUMDAT.
To do a calculation by hand, you need to know the FWHM (full width at half maximum) angle and the lumen output of your light source. Then calculate the illuminated area. Use the formula for the base area of a cone (you can find that at wikipedia) from the FWHM and the height of your room. Then divide the lumen output of your source by the area and reduce that value by roughly 30%. Then you get the illuminance in lux. If you install more than one of these sources close to each other, you can simply add up these values. 200 lx are fairly enough for recreational areas. For more challenging tasks like drawing, you need 500 lx.
I suggest looking for lamps with a beam angle not bigger than 30°, perhaps even 15° is necessary.
One thing to consider if light sources are far away is the formation of very sharp shadows. And with a multiple of distant light sources the formation of multiple shadows, which can be quite annoying. This can be avoided by a flat light source. But most flat light sources have a wide angle making it difficult to bring the light there where you need it.
One way or another this will be some sort of tradeoff.