How can one make up prayers, the number or days of which are unknown, that were performed without ritual purity?

As long as you have not mentioned the number of prayers you performed without purification, you must be cautious in making them up. So, pray what you predominantly believe to be the amount of prayers you owe. This is done by praying each day, in addition to the five current obligatory prayers, what you are able to at any hour of the night or day, and continue doing so until you are certain or predominantly believe that you have made up what you owe.