This paper introduces a high-power broadband X-band klystron amplifier that achieves an output power of 157 MW and a 3 dB operating bandwidth of 6.7%. The amplifier employs an explosive emission diode with high impedance, which balances the requirements of high power and broadband. The multi-gap input and output cavities are designed to operate at two different longitudinal modes within the operating frequency band, resulting in a flat absorption rate and output efficiency. Moreover, an eight-stage stagger-tuned bunching section is implemented to achieve a uniform fundamental harmonic current modulation depth across the frequency band. Simulation results indicate that when the diode voltage is 550 kV and the beam current is 550 A, the amplifier can achieve a maximum power of 157 MW with an efficiency of 51.9% at the central frequency of 9.8 GHz. Furthermore, within the 3 dB operating bandwidth of 6.7% (670 MHz), the power output remains higher than 80 MW. This novel klystron amplifier structure exhibits exceptional performance in terms of bandwidth, power, and efficiency, thus validating the effectiveness of the design in enhancing bandwidth and providing a solid foundation for the broadband design of high-power microwave sources.