you raise, and I wish I could give you a definitive answer, but there isn't one.
The supply of goods and services to consumers regulations (which apply here) state that a supplier has the right to attmept a repair in such cases, and that the repair must be completed within a reasonable time. The 'reasonable time' isn't defined, and rightly so - some items may take longer than others to repair, due to circumstances beyond the retailer's control. Each case is considered on its merits, if and when it comes to a court action.
In your particular case my own opinion is that 3 weeks might be consided a fair period to wait, but once you get to a month you've waited too long. If you were told that the work would take between ten and fourteen days you have a good reason to be annoyed, and I suggest that you write to Argos and tell them that your patience is wearing a little thin. Ask them to give you a firm completion date, and tell them that unless you receive a satisfactory response by return you'll approach Trading Standards for assistance.