Why do you only use potassium iodide for 10 days following radiation exposure?

You don't. One should take FDA approved potassium iodide (KI) for as long as radioactive iodine from a nuclear fission event (nuclear bomb or nuclear reactor only) is present, or soon will be present, in your area. Radioactive Iodine (I-131), which potassium iodide protects against, has a half-life of 8.3 days. Meaning it becomes half as radioactive every 8.3 days. So in 30 days the iodine isotope has became approximately 94% less radioactive. This level is not considered health threatening. When first FDA approved in 1982, the gov't believed that evacuation from contaminated areas, in combination with the quick half-life of radioactive iodine, would mean only a small supply would be needed for a given nuclear event. The FDA picked a two week supply (14 130mg tablets) for the original manufacturers (Anbex and Carter Wallace Labs) only because it was a nice round number. The original packaging use to say, "Take for 10 days or as directed by local health officials". In 2004 it changed to "use as directed by public officials in the event of a nuclear radiation emergency". One would likely be forced to evacuate a radioactive iodine affected area long before they used up all 14 adult doses in a single pack of FDA iOSat Potassium Iodide. This is why the FDA requires each tablet to be individually foilsealed. So if you use a few pills for a given event, the rest would stay fresh for use in the future.

Source: Nukepills.com