I got a bit LED obsessed and started using them everywhere i could.
One thing I came to realise is that they save allot of electricity but do not depend on them as a primary light source(in some places). They lack the luminance of a halogen or florescent tube.. they are getting better though. That said..
Do not bother making your own arrays- waste of time and very prone to failure. You can buy led strips that when one fails only a segment will fail and not the entire array.(well the array within the entire strip- earlier version the entire things fell over what an annoyance)
On ebay (pretty cheap)
You can also look at dx.com or seeedstudio.com-sehnzen4u; they used to be cheap here but it seems they quadrupled in price and ebay fell.. they just do not know how much to charge for them so be carefull not to get robbed in day light.. look around..

So in reply to your question:
Keep the cables you have but the transformer are most likely not going to be any good(you said you had halogen) most main stream halogens run of 12volts.. but AC so this will cause flickering in the LED.. and reduce live they claim.
So you get rid of those and buy some 12volt DC power supplies BUT you need to pay attention to how many amperes they can provide. Ebay once again you could find 4Amp 12 volt and thay should be more than enough to supply up to 6~8 metres strips no problemo.
How to know how much amperes?
You need to calculate the collective Wattage that will run on the one adapter. So lets say you get a 5metre led strip = 25 watts (typically 5W per metre (30LED per metre))
- 25Watts = 2A
- 25w / 12volt = amp
or in other words
- 4A = 48Watts
- 12volt * 4amp = watts
If you put to many LEDs for 1 power supply to handle they will dim.. LED's are curious little things and do not overload the power supply because they can adapt to lower power, without side effects to the power supply. A light bulb demands that there be enough power if not.. then it causes issues.
Some advice
I put over 20 metres in my house in various places. In my kitchen i removed the old dingy under cabinet florescent tubes and run the entire length of the cabinet.It is actually pretty good light as it about 40CM from the table top.


And under my aquarium for sump lighting(used an offcuts I had lying around about 30CM)and also other side off my kitchens under cabinet LED lighting...

Led lights of any type that you want to replace in the ceiling is the issue.. you would need to triple up on the sockets to get some good enough light.. otherwise it makes it seem like the room is misty.. even thought the light is on..

If you only want to use 1 power supply then pop into your computer store and ask for 550Watt power supply (the smallest ones now a days and they are cheap)


What you are interested in there is the sata molex cables. Each one is connected to a serate phase within the power supply. there are 3 phases usually and 1 phase for CPU (ATX connector with all the wires)
Each phase is rated on the power supply 12v@5A 5v@20A for example.. so there you have 12volts with 60Watts capability. Computer supplies are nice because they offer stable, clean power so the LED life are increased thanks to that. It has overload protection(but LED's do not cause overload like I mentioned earlier.. but a short circuit will) so it has short circuit protection too in case something off happens (like LED outside, water or cat taking a bite into the strip .. who know.. cats are dumb) Obviolsy computer shop will stock "GAMING" power supplies at the price of an arm and leg.. go fo the cheapest ones or ask for a used one even.. usually laods lying around in the workshop.
FAILURE
Yikes.. things break! yes they do. I have seen LED's introduced into many places around the world.. the funniest must have been a local council in South Africa that replaced all the traffic lights with LED arrays.. back in 2006. They were amazing- nice and bright and promised a 60%~90% decrease in traffic light power consumption(old 60watt bulbs) BUT.. after 6 months they started failing.. 1 bad LED killed the entire bulb.. oooops. they started melting.. ooops. sometimes only 1 led lit up? huh.. hahah.
That is why i recommend to buy pre made strips.. Look it also happened to me.. why? I do not know.. But its not an issue because...

I can cut that segment out and replace it.. but I just cannot be bothered.. :-)
GOOD LUCK