Generalized nerves of monads

One interpretation of the Kleisli construction (given by Miranda and related to work of Par'e) is as a nerve sending a monad $P$ to the Kleisli double category of $P$. In this paper we find more general nerve constructions on the 2-categories of monads, which also give fully faithful nerve 2-functors $\mathbf{M}\mathbf{n}\mathbf{d}\left(\mathcal{K}\right)\to \mathbf{D}\mathbf{b}\mathbf{l}$.