Welcome to Apug!
Maybe you mix up things as you show us the front of the camera.
The LED at the front (as with other SLRs) is to indicate the self-timer being in action. (It only is to flash, when the self timer fucnction has been chosen and the release button having been pushed.)
In your case, after selecting "P" at the mode switch, push the release half way, andf at least one of the LEDs projected into the viewfinder should light up.
If this does not do so:
-) check batteries for right orientation
-) check batteries themselves resp. get fresh ones
-) check batterie contacts, they must have a silver shine when new
--) if there is salt incrustation, ask again
--) if there is other kind of coating, use a glass fibre eraser on the contact parts, if accessible
If all this is in vain, then the fault is at the circuit within the camera and you better get yourself another sample.