How to make LC circuit just using Capacitor and Inductor?
I assume you try to get use of the fact that in source free RLC circuit energy will flow back and forth between capacitance (C) and inductance (L) and therefore will blink your led. Well, this is not going to work because led is diode and diode as you may know conducts current in only one direction. So what may happen at best is your diode will slowly go off. However your did not limit led current so it may already be dead.

Below circuit on the other hand will blink the leds for you with frequency determined by RC time constant:



How does it work? Assuming capacitor were discharged and left transistor were ON first, the left bulb will be ON while right OFF. But when left trasistor is ON it also cherges its capacitor and when it is full charged it turn right tranistor ON. When this happens the base of left transistor goes negative due to right hand capactor discharge and therefor now only left bulb lights. The whole process repeads indefinetely.