I did say to replace the pulley, but if the 'clutch' has been used, you probably need to change the compressor!
The first thing to do is check for refrigerant pressure with VCDS. Low gas pressure is probably the main reason for failure. The A/C will not function without sufficient pressure and in this case the fan will not run continually when the A/C is switched on. If VCDS tells you that there is no pressure, don't assume that that is correct because the pressure sensor may have failed! So the next step would be to get the pressure checked with a gauge. If this shows zero pressure, get the system filled with refrigerant and green dye; points of leakage will show up under a UV light (an LED light also works well). It is not advisible to take the alternative route of dismantling the system without checking the system with a gauge. The most likely point of leakage is probably the condenser (in front of the radiator), especially if it is original. Replacing the condenser involves removing the front bumper. There are also nine O-rings in the circuit which might be a source of leakage; the two on the compressor are not very accessible and might involve lowering the compressor. It is also a good idea to replace the drier if this is original.
RAB