Yes I also wondered about why this is on the Sussex board instead of the one for Kent?
Best suggestion is that you reveal to us the mother's maiden name on the birth cert you have, which I presume was for a birth registered in Greenwich in the September quarter of 1906. Hopefully we can use the details on it to find the mother's marriage to Frederick Hurley, and from that you/we may then be able to find further children of theirs. Catherine's parents or any brothers or sisters may still be together or with relatives, so using this back door approach may help find the family.
Given the Greenwich location, an immediate possible could be Frederick Charles Hurley who married either Elizabeth Gorrod or Florence White (March quarter of 1905) in the nearby district of Woolwich.
Unfortunately, at this point we also have some controversy, for looking back at the identical link you have left on the Middlesex section of Rootschat, I note that Eyesee has replied there saying:
"Looking at the baptisms for Deptford St Paul, which is in the Greenwich area, there is the following:
Catherine, d/o Augustus & Caroline HURLEY, 34 Catherine Grove, General dealer, born 1906 Jun 10.
The birth date given on the baptism is identical to the date you have given, so was Catherine's father Frederic or Augustus?
Eyesee then adds ..... There was another child to this couple in 1904, Charles Edward and the father is given as William Augustus. Looks like the birth certificate is the first thing on the list, to confirm parents.
There is obviously an error somewhere for the parents on birth cert you have and those on the baptism details Eyesee found, do not match, so one may have been falsified to cover some family indiscression. By starting with a more detailed inspection of the birth certificate in your posession, your findings may eventually solve the mystery as to why your Nana Catherine ended up in an institution.
Roy G