Cables on LED mirror
BROWN and RED are PROBABLY +ve in each case:

Use Ohmmeter with 9V battery.
Set to low Ohms range.
Place probes across LED leads
Brown LED lead is PROBABLY +ve but may not be.
Blue  LED lead is PROBABLY -ve but may not be.
In one of two orientations LED should glow very very very dimly.
For that connection - determine polarity of meter leads (red MAY NOT be positive in this mode) 

If no light either way.
Obtain 1000 Ohm resistor.  

Connect battery black to LED blue
VIA 1000 OHM RESISTOR connect Battery brown to LED brown.
Does LED light very dimly?
If not, reverse battery connections and still using series 1000 ohm resistor try again.
If LED lights this way all is well.

If LED lights neither way, find someone local to help - but one or other way should work.

IF LED never worked it may be dead,

MOST LIKELY:

Battery red brown via switch to LED brown.
Battery black to LED brown.