Let's say a modern man is transported to a world that is roughly medieval and modeled very clearly off of medieval Europe. He manages to impress the son of a king and eventually gets the king to hear out his 'crazy' theories. The king is willing to experiment with sanitation, but is not certain he believes in it and is only willing to commit so much effort to it.
The modern time traveler explains the basics of germ theory and that most medieval diseases were transmitted via drinking water. In addition to explaining the advantage of boiling drinking water he suggests a sewage system. The problem is he doesn't know how to make one. He has a vague idea that modern sewers use filtration, chemicals, and UV light to clean our sewage, of those only the first seems remotely plausible given the tech available but he doesn't know much more about it.
So he suggests that a sewage system can be made to collect waste and drop it off down river of the city so at least they won't be drinking their own waste. He is far less confident about the ability to filter and clean out waste but he vaguely suggests he thinks they may have used sand and/or stones to filter waste before and perhaps they could figure out some way to do that.
Finally he happens to recall a bit about Miasma theory, the flawed theory that 'bad air' makes one sick, and that the first sewage systems were built based off of this theory. Thus while he does clarify that stench is more our bodies system of warning us of stuff that may make us sick rather then the source of the sickness he does suggest a general rule of thumb that if a sewer system keeps the stink of human waste away from the human nose it's probably protecting from disease.
Given this very vague and limited explanation the king pulls together his top experts and asks "think you can do something based off of all this nonsense the supposed future knowledge stranger claims will help? Also I'm only toying with this idea to humor my son's belief in the loon and the potential to maybe make our cities not reek so much, so how cheap could you make this whole sewer idea?"
My question is basically can an effective sewer be made given such limited description at a cost low enough that a king not entirely sold on the idea would be willing to sign off on it? If so can you give me a rough idea of how significant the cost would seem compared to the available budget of such a king and how much of a difference would it make in spread of disease to a city that built one? Would the results be noticeable enough to prove the efficacy of germ theory and/or sewers to the skeptical king?
Less relevant, but related, question would be where the fist sewer would be built. Ideally it would be in the capital, but would a modern already built congested capital city be able to fit in the new construction and sewage system? If not what type of city might be an easier place for a first experimental sewer to be built?
Please note I realize that pre-medieval sewers existed in our world, but this world isn't exactly identical to ours. Either sewers were not discovered in this part of the world or they were forgotten, much as roman sewers were neglected. The point is there is no historical example of a fully functional sewer available to draw ideas off of.