You can easily test a single LED with a lithium button cell and a 1k Ohm Resistor. Of course you can also use any other voltage source with app. 3.2V instead of the button cell. If the LEDs are broken then de-soldering is quite easy with a hot air de-soldering gun. The tricky part is replacing the LED. You need to find a type with matching brightness and color and re-soldering requires a good eye and proper equipment.
There is a cover at the bottom which should be removed and another one in the frontal ring just below the PHILIPS characters. Below that are 2 screws and if you unscrew those you can take off the frontal ring which reveals some more screws. Unscrewing those lets you take of the front diffusor and gives you access to four bigger screws. If you take those off, then there are still two hooks at around 4 and 8 o'clock which keep the front connected to the back. It is a little fumbly to get these off but I managed it by gently levering with a screw driver and nothing broke apart.