The use of networked sensors or humans-in-the-loop for measuring outputs of dynamical systems inevitably leads to the introduction of time-varying measurement delays. The presence of such delays can lead to instability or severe degradation of system performance. In this paper, linear matrix inequality-based sufficient conditions are proposed for the design of state and unknown input observers informed by delayed measurements for a class of nonlinear systems, where the nonlinearities are characterized by incremental multiplier matrices. The proposed observer is guaranteed to perform at specified operational levels in the presence of unknown exogenous inputs acting on the states and measurement outputs. Sufficient conditions are also provided for the estimation of these unknown inputs to a specified degree of accuracy. The effectiveness of the proposed approach is demonstrated via a numerical example.