In this research, we demonstrate a high-sensitivity integrated silicon nitride long period grating (LPG) refractometer based on a rib waveguide with sinusoidally modulated width. While integrated LPG architectures typically achieve ultrahigh sensitivity only over a narrow optical bandwidth using a phase-matching turning-point optimization technique, our sensor exhibits a very high refractometric sensitivity that was designed to remain constant over a broad operational optical spectral bandwidth. The proposed design method relies on multi-modal dispersion tailoring that consists of homogenizing the spectral behaviors of both group and effective indices of the coupling modes. Experimental results are in agreement with numerical simulations, demonstrating not only a sensitivity reaching 11,500 nm/RIU but, more significantly, also that this sensitivity remains almost constant over a broad spectral range of at least 100 nm around 1550 nm. Additional advantages of the proposed sensor architecture encompass a low temperature sensitivity, down to -0.15 nm/K, and simplicity of the fabrication process. These results demonstrate the feasibility of chip-scale photonic integration to achieve both high sensitivity and large dynamic range of the proposed refractometer.