I think some of the information you have been given has confused you a bit.

Sandhill Farm is quite separate to the workhouse/hospital. I imagine that 'Sandhill' was the name of the estate to which the land upon which Sandhill Farm and the workhouse were built originally belonged.
If you follow the link to the listed building which someone kindly gave you before, that is most definitely Sandhill Farm. If you look at the OS map given on that link and follow Tonbridge Road southwards, you will find the hospital. This is where the workhouse used to be. Once the workhouse closed, the buildings started to be used as a hospital and other buildings were gradually added over the years. Most of the newer buildings on the site have already been demolished to make way for the new hospital. The old workhouse buildings are still there (I used to work in that part of the hospital), but will soon be flattened to make way for more parking spaces. The only part that will remain is the old workhouse chapel which, after local campaigners got it listed status, has been saved.
Here's a link that explains a lot more:
http://www.workhouses.org.uk/index.html?Tonbridge/Tonbridge1881.shtml Now that I have read your first post more carefully, I'm wondering if 'Sandhill' as your birthplace refers to the hospital, as you say your parents' address is elsewhere in Tunbridge Wells. The above link says that the workhouse became a hospital in 1938, so that's only 2 years before you were born.
The 5th paragraph in the link under the heading 'After 1834' expalins that the land purchased to build the workhouse on was at 'Sandhill'. this could explain your birthplace also.
I do hope that this helps
