I downloaded VS 2013 Professional. I created a Silverlight 5 solution. I allowed it to make the web parts. I ran it and it ran fine.
Then I created another solution and did NOT select to create the web parts, but only the Silverlight. I ran it and it said Silverlight was installed and gave me a UI to click to get Silverlight. Which tried to install it and said it was already installed.
I presume there is some problem in using Silverlight when it is running it from a file. Is that correct?