Get rid of the LED.. if you absolutely cannot find a diode (not even in the scrap pile?) at least use a transistor connected as a diode! You can do this with bipolars or Fet types... if you use a bipolar, don't use the base emitter junction though because it will breakdown at around 7V more like a zener diode than a normal one.
Also keep in mind, diodes made like this are not fast, they have recovery times on the order of hundreds of nanoseconds..so it might STILL not be good enough for you, but you can try it!