Over the past years, the idea of the KL property has become increasingly common in the optimisation area. It was used as tool guarantees that every sequence generated by many different descent algorithms enjoys finite length property. When verifying the KL property, one needs to find a desingularizing function, which is usually considered difficult. In light of this, our paper provides a desingularizing function that can be regarded as the smallest among all potential desingularizing functions. We investigate continuity, Lipschitz continuity, differentiability and subdifferentiability properties of this function as well as different characterizations of the so-called Kurdyka-Lojasiewicz-Hoffman constant by introducing a new class of functions with nonsmooth moderate behaviour. Based on these attributes, we provide a convergence analysis and an estimation of the convergence rate of an inexact descent methods for convex differentiable functions that covers a wide class of gradient descent methods. We also delves the complexity property of this method and its relationship with λ-one-dimensional worst-case proximal sequences.