What kind of desk lamp is it? If it's a flourescent bulb type, there is probably a starter module that can be replaced.

If it's an incandescent type, it could be the switch or power cord--when transformers go, the wire opens up _somewhere_ in the transformer. Uncoiling hundreds of feet of tiny wire to attempt to repair a break is nigh impossible. If you can find the part number on the transformer you can usually google a replacement.