Why is GetKeyDown returning false when provided with string from .inputString?

Apparently, internally Input.inputString is updated after GetKeyDown is executed, thus keyDown is empty when you read Input.GetKeyDown. For what do you need this? Tell us, and maybe someone can suggest you another approach.
NOTE: Ooops! You've already said what's your intention! Let me think for a while and soon I'll return with something.

EDITED: inputString is only updated one frame after the key is actually pressed, thus using GetKey functions is somewhat complicated. But there's a simple solution for this case: when inputString isn't empty, compare its value to the last key pressed: if it's the same, check the time elapsed since last key - if it's lower than some predefined limit, you have a double click:

 var tDoubleKey: float = 0.25;
 private var lastKey: String;
 private var timer: float;

 function Update(){
     if (timer > 0){ // if timer > 0, decrement it
         timer -= Time.deltaTime;
     }
     var key = Input.inputString; 
     if (key != ""){ // if any key pressed check double click:
         // if key = last key and double click time not elapsed...
         if (key == lastKey && timer > 0){
             // we've got a double click!
             print(key+" double-pressed");
         } else { // otherwise update lastKey and restart timer
             lastKey = key;
             timer = tDoubleKey;
         }
     }
 }

You could use OnGUI to detect keyinput, however, like i said somewhere here, OnGUI wraps the OS input and when you hold down a key it will be repeated automatically at the repeat-rate configured in your OS. So you have to detect both, KeyDown and KeyUp

 var key : KeyCode;
 var releasedOnce = false;
 
 function OnGUI()
 {
     var e = Event.current;
     if (e.type == EventType.KeyDown)
     {
         if (!releasedOnce)
         {
             key = e.keyCode;
         }
         else if (key == e.keyCode)
         {
             releasedOnce = false;
             Debug.Log("You pressed " + key + " two times in a row");
         }
         else
         {
             releasedOnce = false;
             key = e.keyCode;
         }
     }
     else if (e.type == EventType.KeyUp && e.keyCode == key)
     {
         releasedOnce = true;
     }
 }

Can't guarantee that there are no syntax errors. I wrote this from scratch and usually i don't use UnityScript.

edit
This does also work:

 var key : String = "";
 
 function Update()
 {
     var keyDown = Input.inputString;   
     if (keyDown != String.Empty)
     {
         key = keyDown;
     }
     if(key != "" && Input.GetKeyDown(key))
     {
         Debug.Log(key+ " pressed twice ");
         key = "";
     }
 }

The problem with InputString is that it is filled one frame after the actual keypress occurrs AND is only set for one frame. It's automatically cleared after that frame.