I know the money thing can be difficult--been there done that. Having said that, in your place I would pay for the Dr. appointment. I know it lets W off the hook for her responsibility, but it is for your D.
You might call the Dr. office and give them a credit card number. That way you wouldn't have to give W any cash. Or, you could take D to the Dr. yourself--that way you get to assess if she is really sick, or if W is just jerking your chain to get you to fork over some cash.
