What would cause a led light to turn off after one second?
We've seen the "one second" thing before.  The issue was an electronic switch (dimmer, motion sensor, etc.) which used a 2-wire installation - it's designed to work in a traditional "switch loop" where it has no access to neutral.  It sits in series with the light bulb.  When off, it exploits the fact that incandescent bulbs have very low impedance when unlit. When on, it's counting on a lot of current flow through the incandescent.  These devices do not play well with CFL/LED. 

Perhaps it's a dimmer and you don't even realize it. Some dimmers look exactly like plain light switches, just with a range of positions. 

Upshot: check out the switch, replace.  Switches (which aren't dimmers) are about a buck.