叫婆婆來我家住看住姐姐 - you create trouble for yourself and 婆婆 ->> 要看住
If she is not performing, there is little difference with her for that month or not. It only makes you feel less psychologically unbalanced because you make her work. In fact, you need to spend efforts and energy on managing her and it bothers 婆婆 as well. If you count your efforts and energy as well as 婆婆's + the risk of potential damages of your household properties, this is definitely more than $3580. Hence, why don't you give her $$ - problem free and no worry of being sued 無理解僱.
This is unfair to us as employers but this it the game rules.