In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic problems by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$. We present a multigrid and a two-level domain decomposition approach in two or three space dimensions, which are steered by their respective a~posteriori estimators of the algebraic error. First, we extend the results of [A. Miraçi, J. Papež, and M. Vohralík, SIAM J. Sci. Comput. 43 (2021), S117–S145] to the mixed finite element setting. Extending the multigrid procedure itself is rather natural. To obtain analogous theoretical results, however, a multilevel stable decomposition of the velocity space is needed. In two space dimensions, we can treat the velocity space as the curl of a stream-function space, for which the previous results apply. In three space dimensions, we design a novel stable decomposition by combining a one-level high-order local stable decomposition of [Chaumont-Frelet and Vohralík, SIAM J. Numer. Anal. 61 (2023), 1783--1818] and a multilevel lowest-order stable decomposition of [Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297–332]. This allows us to prove that our multigrid solver contracts the algebraic error at each iteration and, simultaneously, that the associated a~posteriori estimator is efficient. A $p$-robust contraction is shown in two space dimensions. Next, we use this multilevel methodology to define a two-level domain decomposition method where the subdomains consist of overlapping patches of coarse-level elements sharing a common coarse-level vertex. We again establish a ($p$-robust) contraction of the solver and efficiency of the a~posteriori estimator. Numerical results presented both for the multigrid approach and the domain decomposition method confirm the theoretical findings.