Blue Toon,
Assuming you haven't sorted this in the last 3 weeks.
It sounds like the transformer and it is very easy to replace.
Before you start changing things, first check the total wattage of the bulbs does not exceed the maximum rating of the transformer. If it does then the transformer will not work!
If all is in order and you still suspect the transformer is faulty the purchase a new transformer of the correct rating.
Isolate the supply to the circuit and, using a mains tester, CHECK THE CUIRCIT IS DEAD.
Once you are sure the power is off, it is simply a case of disconnecting the 2 cables, (240v mains cable in & 12v cable out), of the transformer and substituting the new transformer before reconnecting the cables. Check that the transformer is the correct way round and the 240v in goes to the feed and the 12v out goes to the load.
Good Luck.
Paul.