Starting from a fixed point relation, we construct very fast iterative methods of Ostrowski-root's type for the simultaneous inclusion of all multiple zeros of a polynomial. The proposed methods possess a great computational efficiency since the acceleration of the convergence is attained with only a few additional calculations. Using the concept of the $R$-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step method with Schröder's and Halley's corrections under computationally verifiable initial conditions. Further acceleration is attained by the Gauss-Seidel approach (single-step mode). Numerical examples are given to demonstrate properties of the proposed inclusion methods.