Why won't this 555 timer flashing circuit work?
Looks okay to me. Try a fresh battery (almost 100% sure that’s your immediate issue), and increase the series resistor to the LED to draw less current (try 2K or so), at least temporarily. I also can’t see the color code clearly on the existing part.

You seem to have two 1uF caps, put the remaining one across the battery, observing polarity, of course. 

The ancient bipolar 555s draw a large pulse of current when switching, so a bypass cap is desirable.