Circuit for childrens "electric" toy/game
This sounds like one transistor to me - pull-up a large delay cap through the contact and a small resistor, and connect it to the base of a NPN transistor with a larger resistor.  

Have the NPN transistor turn on the LED.

When the contact is closed, the cap will be charged and the base of the transistor will be supplied through the contact.  When the contact opens the cap will supply the transistor for a while.