As above, it'd be worth contacting your local CAB to discuss it with them. Also check your tenancy contract, the original listing, and the inventory carefully.
If the property was advertised with white goods and these are listed on the inventory though they are his responsibility to maintain (assuming you're in the UK) to the best of my knowledge. Note that if the washing machine was purchased by yourself or brought by yourself then it's yours and your responsibility. If it was included in the flat but there's no real evidence of this (e.g. not listed on inventory or letting listing) then it may be more challenging to prove your case if the above is correct.
Once you speak to the CAB, if they advise similar to the above, ask if they can give you a written statement to that effect and give a copy (copy! not the original) to the landlord and point it out. I suspect if true the landlord would realise that it'll be cheaper to just fix it than have to go to court, pay fees, then fix it anyway.