Ian Jelf wrote:

I've heard a story, second hand and from a "Non Railway" person,
which I didn't fully understand, that there was once a station
*somewhere* (not KC) which had a Platform 9 3/4 and that that was the
inspiration for JK Rowling.

Anyone else here know?

I don't, but there are quite a few houses around which have half-numbers.
I've never seen a three-quarters though.

There are bound to be some platform signs which bear a three and a four,
possibly with a slash between them. (I've seen a sign in a hospital marked
"Wards e1-e10", which has "Old McDonald had a farm" graffitied on it.)