I agree wholeheartedly that a pet is much more than "just a pet" - he or she can be something that is worth more than any amount of therapy to someone with dementia. To remove the pet is not only cruel, but can most assuredly be counterproductive to their mental and physical health and well being.
Talk to the facility manager first and see if they can hold off for a short amount of time so you can do the following;
1) Clean up the apartment floor with odor eliminating carpet and hardwood/vinyl cleaner. If you aren't able to physically do so yourself, there are many cleaning agencies that can do so for you.
2) Call volunteer programs for free pet walking services - Senior Volunteer programs, school organizations, sororities, fraternaties, girl scouts, boy scouts, etc. - and solicit them to take the animal for regular walks. (Who knows - you might unknowingly find someone who will be a wonderful addition to your care giving circle).
3) Take the dog to the vet and make sure he/she has no health problems that might make "going" more frequent.
I have been taking care of my foster mom for several months now. We live in the country and a stray dog just happened up on the doorstep one day. I was worried about the "additional" mouth to feed, having it jump up on her, etc., etc., etc. I was thinking, okay, how do I get rid of this dog, who do I call, when I saw a wonderful transformation on her face. A light I hadn't seen in so long shone in her eyes as she spoke sweetly to the abandoned stranger. It was love at first sight for both of them.
A visit to the vet to make certain we had no health problems and a trip to the store for dog food, treats, collar, etc. resulted in a wonderful addition to our family... not to mention a measure of safety for us and a comfort to know if Nanny ever wanders, she will have an angel right beside her.
Good luck and God Bless!