Hi Bart.
This should be pretty easy. This is a 3mm LED. Order a white 3mm LED (I like warm white, or sometimes called "sunny white," instead of bright white, but you have a choice) from your favorite model electronics supplier - Streamlined Backshop in Indiana has them, All Electronics, Richmond Controls, etc. They aren't hard to find.
Now - notice the orientation of the LED on the board. You'll see two elements inside the LED housing - a small one and a large one. The small element (on the right side of the LED in your photo) is the anode, or positive connection; the large one is the cathode, or negative connection. But all you really need to do is note the orientation of those two parts and remember it - take a photo, mark it with a marker or whatever.
Now the "proper" way to replace the LED would be to desolder it by heating the two solder pads holding the leads and cleaning off the solder, loosening the LED leads, which then allows you to pull the LED out from the pad holes. But this is easier said than done unless you have some sort of desoldering equipment, like a desoldering braid or vacuum bulb. So what I do is simply cut the current LED off at it's base, then solder the leads of the new LED to the remaining leads of the old LED. REMEMBER THE ORIENTATION - the new LED should have the same anode/cathode orientation as the old one. [Another way to do this without desoldering equipment is to cut the old LED, then to desolder the remaining leads one at a time, which is much easier than trying to desolder both at once so you can remove the old LED whole. Simply heat the solder blob on the pad and use a pair of tweezers or needle-nose pliers to remove the remainder of the old LED lead from the hole when the solder has liquified. You can then clean out the pad hole with a small drill bit).
That's it - you're done. It sounds harder than it is - shouldn't take you more than five minutes once your soldering iron is hot.
John C.