We have been to Whitby three times since July last year. The first time we went we drove up the coast to Sandsend and happened to see a sign saying "Teas" on a driveway - so we drove up, past a building site, to a lovely hotel called Raithwaite Hall. It is a large, new, stone built hotel with the most wonderful grounds at the back. It was quite pleasant summer day so we sat on the terrace with our teas and thought "we really like it here". We have been back twice since then and stayed at the hotel (both on deals through Groupon) - it's really lovely and the food is excellent. All locally produced stuff. I know your accommodation is sorted out, but if you are looking for somewhere to have afternoon tea (either the proper one, or even just a cup), I would recommend it.