Word puzzles usually include all the information we need to solve it. The non-contagious is there to let us know that each monk either has the disease or doesn't have it, and that's not going to change. "No mirror" means that they can't look at themselves; if they could look into a glass of water or a spoon, there would have been something in the puzzle to lead us in that direction. The wording of the puzzle tells us that the only way they can find out if they have the disease is by looking at the other monks once a day.