Wiring a 74LS259 (addressable latch)
You don't show the connections on your schematic, but I'll assume you actually have them wired up to the switches you mention.
It looks as if you have C and E the wrong way round - they are active low inputs, so you need to tie C to +5V, and E to GND.
Do not wire the unused outputs directly to a power rail (e.g. +5V or Ground) as this could cause a direct short if the logic level is set to the opposite rail (e.g if you have the output wired to ground, and you set it high, then a lot of current will flow and potentially damage your chip) 

So set your data pin D to the desired logic level for your output (e.g. high to light your LED), then A0 and C high, then pull E low. Q1 should light up (if it doesn't check the LED is the right way round, power is present, etc)

Also, if you only have a single supply like is shown, then the low side is generally referred to as ground, not -5V.

The bottom section is the relevant table in the datasheet for addressable latch mode:

 

You can see you must keep the clear pin high (tied to +5V) otherwise it will keep the outputs low (i.e. "cleared")

Also note this bit (the mention of Enable and Clear being active low - denoted by the bar above the letter):