Well a little bit of investigation answers some questions...and raises some about the editing of the show!
I've found the post the two girls mother wrote...its actually just a brief paragraph. The most important thing to the above discussion is the mother (and therefore Danny and the production team) already knew as a fact that Miss Howard had lived at that address. The belated discovery of these details on the show is indeed a bit of theatrics.
So it would appear they assumed the identity of the ghost based on knowledge of who had lived there.
How Kate, the later occupant, knew the name may remain a mystery or may be surmised to be a case of the story of visits from Miss Howard was passed on between seller and buyer (Kate's parents) when the place changed ownership. (Edit: an even more disappointing option....we only have Kate's word that she "always knew" the ghost was called Miss Howard. It may be more a case of having discovered the matching post from the previous owner which identified the apparition by that name she convinced herself that the name in it was familiar/seemed right)
In terms of who else experienced phenomena, the family dog apparently aware of a presence and the parents frequently heard phantom footsteps.
Here is her account in its entirety, posted in 2008.
"I lived at 65 High Street from 1964 to 1971. The two little figures on the left of the photo are my two daughters, Lisa and Jane. We shared our house with a ghost who, we believe, was Miss Howard, who had previously lived there. Her father owned the baker's shop next door and had built the two houses next to it. Miss Howard 'appeared' to both my daughters at different times, when they were 18 months old and 3 years old, and I think that, at that young age, they were not capable of inventing such a story. Our dog was also aware of her presence, and over the years we were there, we frequently heard her walking about, along the landing and up and down the stairs. I've often wondered if any subsequent families living in the house has had similar experiences. My name in those days was Joyce Thorn, if anyone remembers me or my family, I would love to hear from them."
https://www.francisfrith.com/uk/melbourn/a-ghost-in-melbourn-high-streeet_memory-46271