After I watched it again I understand it differently than the first time. I think “on your slip” means she fell on top of her slip when she landed on the floor.
Nearlynapping is probably correct since she’s ironing and doing laundry. It might possibly be a pillowcase, but that would be less common. For the joke to work well, though, there should have been a previous reference - either verbal or visual - to a slip so that the tinman’s statement has some context.
And just to confuse things, in another context “slipped on a slip” could mean that she put on a slip.