Highly sensitive microwave devices that are operational at room temperature are important for high-speed multiplex telecommunications. Quantum devices such as superconducting bolometers possess high performance but work only at low temperature. On the other hand, semiconductor devices, although enabling high-speed operation at room temperature, have poor signal-to-noise ratios. In this regard, the demonstration of a diode based on spin-torque-induced ferromagnetic resonance between nanomagnets represented a promising development, even though the rectification output was too small for applications (1.4mVmW(-1)). Here we show that by applying d.c. bias currents to nanomagnets while precisely controlling their magnetization-potential profiles, a much greater radiofrequency detection sensitivity of 12,000mVmW(-1) is achievable at room temperature, exceeding that of semiconductor diode detectors (3,800mVmW(-1)). Theoretical analysis reveals essential roles for nonlinear ferromagnetic resonance, which enhances the signal-to-noise ratio even at room temperature as the size of the magnets decreases.