Need a lead on a lamp part  

cool
The design on these may be a bit outdated,but they are never the less cool. Like all italian lighting, there is a time and a place for these. I am jealous of your find. I am certain that there aren't many of those around anymore. And maybe the monetary value is not too high, but I think you should save these for sure.
What you need is not too complicated, especially if you have basic handy skills.
you will need something like this: http://www.lightbulbsurplus.com/index.php?main_page=product_info&product...
Or I found this for you: http://mymanthemaker.blogspot.com/2013/04/ikea-12v-halogen-lamp-fix.html
I think you will have a very hard time finding original replacements from Luceplan, and if by some crazy hance they still have them in stock they would probably charge a lot.
Can you post a pic of the plug in terminal(the one that plugs intom the transformer?

Yep as said above.
You...
Yep as said above.
You just need a transformer that outputs 12 volts maximum and 20 watts minimum. Should be easy to find. And splice the cord. You might find one that has a secondary plug at the transformer like the original. Then you can splice the cord inside the lamp where it is hidden and run the new cord to the new transformer.
And if the transformers are cheap you might get a couple of extras. They don't last forever. Although the higher the wattage rating the longer it will probably last.