I would thank her for taking care of the cat and thank her for letting them know where the missing cat was living the last few month. Also tell them that you've taken your kitty to the vet and he is getting better and putting on some weight. That way the woman who took care of your kitty can feel pretty good that he has returned to a nice home.

Do tell her that next time he visits and stays to give you a call so you can go and pick him up.

If there is a street behind your house, I would highly recommend to turn him into an indoor only cat. I realize it can be very traumatizing for the kitty to not go out but they eventually will get used to it. Safer for him. Not get hit by a car. Not get chomped by a dog racoon, nor get in a serious cat fight. Cats will kill each other if its a tom.