Transformers either work or they don't work and it's quite rare for a transformer to fail and most unusual for a transformer to work sometimes and not other times - I have never seen such a fault.
The chances are you have a poor contact problem. Possibly a broken wire inside the insulation - rare and difficult to diagnose but after 20 years of flexing worth considering.
http://wiki.answers.com/Q/How_do_you_replace_the_halogen_source_in_a_Tizio_desk_lamp
Peccavi
October 2009