Depends. With an Arsenal we should be able to outfit elites with steel equipment immediately(see the turn after Fey Mood for how fast we could have deployed iron starting from "figured it out from the ore", to "functional weapons" by the mid turn.), then propagated it outwards
The next question would be the utility, though.
Is steel such a qualitative leap forward for our armies as Iron was?
Right now, I think heavy armor and the needed production techniques would serve us better. Beforehand, bronze weapons were rare enough that being faster beat being tougher, but if the Khemetri army is like the New Kingdom Army of Egypt, they will all have bronze weapons. Being able to field heavy infantry would no sell that advantage. We would be much slower, but they couldn't actually hurt us and the Carrion Eaters have good odds healing anyone that was injured.
Our existing scale mail may work well enough, I suppose. But I don't know if it is widespread enough. Maybe it's the reason why we managed to hold the line.