The solar irradiance received by the Earth is 1361 $W/m^2$. What is the lunar irradiance received by Earth from the full Moon in the same units? My first thought was to work it out via apparent magnitude but I don't think that would work out.
You can work in magnitudes, but the magnitude scale is logarithmic. Instead you can use luminosity. The moon has a luminosity that is 400,000 times less than the sun, so the maximum theoretical "lunar irradiance" is about 0.0034 watts per square metre.
In practice, the luminosity at the surface is less than this, with Wikipedia quoting "how bright is moonlight" typical values on the Earth's surface of 0.1 lux, equivalent to 0.001 watts of visible light per square metre.