Kate HAYSOM's birth is indeed shown in the FreeBMD site (online, partially transcribed, England & Wales births, marriages, death certificate indexes) at http://freebmd.rootsweb.comhttp://freebmd.rootsweb.com as follows:
Births registered in the July-Sept quarter of 1872 Kate HAYSOM; District= Isle of Wight;Vol 2b, Page 553
As you know her father's name, you probably have this info already. If not, when you buy a copy of the certificate you will find out where on the Isle of Wight she was born, and her mother's maiden name.
There is a marriage listed at the site as well, of a Timothy James HAYSOM (Oct-Dec quarter of 1868; Isle of Wight; 2b 998). There is an Ellen Harriet JACOBS getting married at the same time, and this may well be Kate HAYSOM's mother. You will need to investigate further by getting these certificates. (Let me know if you need to know more about how to do this.)
The chalk cliffs of the Isle of Wight are actually visible from Swanage on a clear day. I'd bet that Timothy - or perhaps his grandfather - came from Swanage, just like most of the other Haysoms at this site.