We present the first measurements of kilowatt laser power with an uncertainty less than 1%. These represent progress toward the most accurate measurements of laser power above 1 kW at 1070 nm wavelength and establish a more precise link between force metrology and laser power metrology. Radiation pressure, or photon momentum, is a relatively new method of non-destructively measuring laser power. We demonstrate how a multiple reflection optical system amplifies the pressure of a kilowatt class laser incoherently to improve the signal to noise ratio in a radiation pressure-based measurement. With 14 incoherent reflections of the laser, we measure a total uncertainty of 0.26% for an input power of 10 kW and 0.46% for an input power of 1 kW at the 95% confidence level. These measurements of absolute power are traceable to the SI kilogram and mark a state-of-the-art improvement in measurement precision by a factor of four.