Well done Gwynant, it is Lady Eleanor's Tower and is a listed building.

Lady Eleanor's Tower, Llanddulas and Rhyd-y-Foel
Location
Strikingly located on a wooded crag, set back from and above the road; approximately 750m W of Gwrych Castle.
History
Early C20; constructed for Countess Dundonald of Gwrych Castle and one of the last of a number of folly towers, gates and similar erections built on the estate from c1819 until the beginning of the first war. Traditionally said to have been built by the grieving countess in 1915 to commemorate the loss of her son at sea.
Exterior
Square tower of two-and-a-half stages; of local limestone rubble on rock foundations. Segmentally-headed window openings to N E and W with access to the S; large corner crenellations give fictive battlements.
Reason for Listing
Included as a striking folly tower on the Gwrych estate, part of this nationally important Picturesque composition.
I hope that your hip operation has been a great success and that you are now free from any pain. I know people that have had it done and been very pleased with the results. We did the walk which was a bit of an exploration too as we visited the caves at Tan Yr Ogof too. Will be posting that asap after our trip now to B & Q. Hope that you have a swift recovery.