Conversion of a Cold Cathode kit to work outside a PC
ADDED

Note that the inverter for a CCFL is a very very specialised device. You cannot replace them with anything else "off the shelf" that is not intended for CCFL use. So, you feed them what they want but don't try to replace the actual inverter. So ...

The nameplate says that 


The input voltage is 12 VDC.
The Wattage required is 2.85 Watt +/- 0.3 Watt. 


So you want at least 2.85 + 0.3 = 3.15 Watt, and a bit more is a very good idea. 4 Watt or 5Watt.

Your 12V power supply says 12VDC , 1A.
 Power = V x I = 12V x 1A = 12 Watt = ample!

Your original referenced pictures showed a PC power supply connector, 



which I assumed connected to the 2 x 2 small plug that you now show.

BUT - does yours have the large flat PC connectors shown (4 wires in a line, red-black-black-yellow wiring)?   If not, the notes below make less sense that they might. 

A problem is a lack of information still.
 You new photos are better BUT we still don't know your whole configuration. 

IF you have a PC power supply connector, see below.

BUT if the sole power connector is the one you show - How does it get power from the PC?



The "easy and obvious" path (2 choices) is to either 


Add a PC 4 pin power supply socket as per your 1st photo to your 12V, 1A supply and plug the CCFL PC plug into it. Or ...
cut off the PC plug to the inverter and solder your power supply to the wiring. +12V = yellow (probably). Ground or -12V = black (probably).


The first choice leaves your inverter wiring untouched so that it can be changed back just by unplugging the power pack and plugging the CCFL system into  the PC if required. PC power sockets are very very very commonly available in most countries in old dead PC's or on PC power supplies  or from a computer shop. This is the best choice if you have enough room.

The second choice is very slightly more compact and eliminates a connector pair but otherwise has few advantages and it means you cannot easily change back.



Note that connecting power in reverse may  destroy the CCFL inverter or the power supply or if you are clever enough, both, but probably not. ie note all voltages colours and connections carefully and draw a diagram of what you intend top do, no matter how simple it may appear. 



OLDER - still worth leaving in:

You should be able to drive this from a suitable mains power supply - that is in fact what the PC power supply is.

Points to consider are


Which volages does it use?
PC connector supplies +5V / ground / ground / +12V.
red / black / black / yellow
You need to determine whether it uses just +12/ground or just +5/ground or +12/+5/ground.
I'd expect probably just +12/ground
The supply has to be "adequate"


It has to make the correct voltage without "sagging" or varying "too much". 
It has to handle startup and restart transient loads which may be heavy compared to average load.. May.



You show a 12V 400 mA suppply . 
 That makes 4.8 Watt nominal.

This page suggests 5ma at 680 Volt for a  300mm tube.
 Power = V x I = 680v x 5 mA = 3.4 Watt.

So your supply would be notionally OK for that tube. But not by a vast margin.
 Startup cu                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            rrent MAY be an issue.
 How well regulated that supply is is unknown.

More soon ...