How would I create such a thing? (Wired lighting)

I want to understand more about this whole "thing" (light addiction) by making a few projects. The first one I would like to try is basically a cabinet light. It needs to light up the space under the bathroom sink. Don't ask why :nana:
The issue will be wiring it up to the wall outlet. I know I can get a wall wart and cut it apart to supply 12V or 9V AC or DC.
I'm thinking 5 to 10 of those 5mm white LEDs on DX. See here for specs. If they don't "break through" a sheet of paper acting as a diffuser I might need to go with the bargain bin Cree/SSC/Luxeon LEDs.
So tell me how to wire them up, what resistors, ICs, transistors, capacitors and power adapter I need. I'll worry about the directivity and fixturing.
Or tell me where to find what I need to know to think it up myself.

ok, ill dive in!
I think you need to experiment really, ie try different methods to see how much light you actually need, and what method will supply this, of course you might get lucky first time, and be happy with your first effort.
SO in this fashion, what I would do, is get some pre-made led strips runnng off 12v to begin with. DX recently brought some in, and there are lots of other sources, dont know who makes the good ones, so it will be best to ask for advice from other members for these.
Then if you need more light you might wanna go the Power LED ie cree etc route, and for that you either need a resistor, or a driver circuit, or if you plan it well (ie your V, Watt requirements) you might be able to direct drive from a suitable PSU (eg you can direct drive a Cree from spme cell phone chargers, although their total output voltage is higher than the LED limit, they are current limited, so they are great for driving single leds). Which one, depends on many factors, eg which leds, how many, supply voltage, etc. You will also need a good heat sink structure. Each of the above factors are discussed in CPF, and also reading a few data sheets will give you a good inderstanding.
Hope this is useful to you. Good luck!