OP - I have been through this with Comet (before they went out of business).
You can of course expect a domestic washing machine to last more than a year.
As Chandler 12 writes above, you can ask Argos to replace or repair the machine. If they refuse, you can take them to the Small Claims Court, or if you paid by Credit Card, you can make a Section 75 claim against the Credit Card company.
Your sales contract is with Argos and it is Argos who you should be corresponding with (unless you go to the Sect. 75 route). Retailers will try to fob you off and tell you to go to the manufacturer - you politely reply that they (in this case Argos) are the ones responsible for making the repair or replacing the machine. They might attempt to get you to pay for an engineer to come out (you've already had one - did he leave a report?) or attempt to get you to ask the manufacturer to send an engineer, again at a cost. Just remind them that it is their responsibility, not yours, to provide an engineer. If you do eventually decide to get one, in order to move things along, find a local one who might charge you £50 rather than using 'their man' who might charge you £150 (it's a tactic to get you to drop the claim).
Which (see posts above) have template letters that you can use, and also tell you the steps that you must take in order to make a claim. You must do Step One, allow Argos time to reply, then Step Two and so on, and then which forms you need to complete if you need to go to court. Always keep copies and always send letters by registered post, so you have proof of delivery.
I ended up at the Small Claims Court, which is just you (you can take a friend or partner), the judge and the representative from the retailer sitting round a table. The judge will have read the papers beforehand and will ask any questions that he/she feels necessary, before making a judgement. My case took about 5 minutes - the judge asked Comet why they were even bothering to defend the case when it was clear cut. They had to pay the refund, plus compensation for my court costs, including postage and fees, and the engineer's fee and if I had pressed the matter, I could have asked for compensation for half a day's wages (time spent completing all of the paperwork, plus the time off work for the hearing).