Does any current flow through C-E of NPN BJT when the base is floating?
Okay, looking at the picture I think you may have the transistor the wrong way round. 

Try turning it round.  

See this picture for reference:



As you can see the collector is on the right with the flat part facing you, so you have the collector connected to the LED in your circuit (if the 2N2222A part you are using has the same pinout)
I got the picture from here. 

EDIT - It's actually a 2N222A, but the above advice still goes as the pinout appears to be the same from the picture posted.  

As Russell mentions the more standard way is to connect the LED to the collector, but your circuit should work if set up correctly.