Your post ignores a very important variable - price.
Who exactly is unwilling to work as domestic help?
Pretty much anyone, at $10-12/hour.
People who can afford to pay $35-40/hour have their houses cleaned by a maid, but there is nothing special about this price point. If we have a lot of un-utilized labor who are willing to clean houses for less than $35/hour, prices should fall.
Since prices have not fallen, we must conclude that our unemployed are unwilling to clean houses except for high wages.
(Or possibly demand for domestic services is completely inelastic, but that's unlikely. I'd certainly have my house cleaned more often if it were cheaper, and I know many people who can't afford it at $35/hour but could at $10/hour.)