Anything that LL does that is "normal" cleaning, is not your responsibility. You are responsible for returning the unit to the condition it was when you moved in, minus normal wear and tear. If the LL chooses to hires services as a matter of course, it is not your responsibility. The LL must also show that this has been done in the past after every previous T to even make the claim of "normal move out cleaning." The LL could have refused to let you move in until after the cleaning was done, the lack cleaning before you moved in is not your responsibility.
To get your money back you will have to sue in Small Claims court. Since it will be a he said she said it will come down to who has the best evidence. Bring before and after photos and any witnesses to the state of the property when you received it(any neighbors or non-related individuals will hold more weight in court. I think you have a good case; the landlord would have to show the place being quite a wreck for witholding $1300. You can sue up to 2x the cost of your security deposit in WA. Google RCW 59.18.280.