If you get a copy of her birth certificate it will give you her full date of birth (and her full middle name), and you can search on the England & Wales death index (on ancestry) for people with that exact date of birth, but if she was adopted as a baby, her first name may well have been changed, so you might not be able to work out which one is her. But if you know for sure that she is dead, maybe the person who told you that she was dead can give you some more details like roughly where and when she died to help narrow it down.

(It sounds at the end of the post as though you have already seen the actual birth certificate though, in which case you should already have her full name and exact date of birth. But I don't know why you would be surprised that the mother's maiden name was on the birth certificate, since all birth certificates have the mother's maiden name on them?)