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You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)