Assuming we have the tools and materials of the late medieval period in Europe (~ 15th century) but are allowed to use modern knowledge of metallurgy, what method/process of steel production would allow the highest quality steel to be produced if the measure of quality was its use in medieval weapons and armour?
It doesn't have to produce industrial quantities of steel, but there should be enough of it to feed a few smithies. Any source of iron available in medieval Europe can be used, so both mined iron ore and bog iron are available, as are any others I may not know of.
Does this change if we add some kind of magic fire to the mix that can achieve temperatures in excess of 2000°C and if so, what process is now the optimal choice?
EDIT: To clarify: I'm not asking about the methods that were actually used and available in the 15th century. I can easily look those up. I am asking which superior methods could still be implemented with 15th century tools and materials assuming the neccessary knowledge of said methods.