There is no exact conversion factor.
The problem is that lux and W/cm^2 are not simply different units, rather they are units for different measurements.
mW/cm^2 is a unit for irradiance (energy falling on a surface) and lux is a photometric unit for illuminance (visible light falling on a surface). The Light Sensor, order code LS-BTA, $59 is optimized for the wavelengths seen as visible light, weighted much like the human eye (see the Light Sensor information sheet). When measuring lux, you are measuring this visible brightness.
On the other hand, W/cm^2 is a measurement of absorbed energy, or irradiance. This may happen from a much broader spectrum.
There is no single conversion factor between lx and W/m^2 as there is a different conversion factor for every wavelength. The spectral composition of light would be required to estimate a conversion factor.
The Pyranometer, order code PYR-BTA, $224 measures irradiance.