There are plenty of places to stay in Kirkby Stephen itself or nearby villages like Ravenstonedale or Brough.
Many accommodation providers in the area are committed to the Walkers are Welcome scheme and have special facilities and services for walkers. If you arrive by public transport your hosts may be able to pick you up from the railway station or bus stop and take you to and from walk starts. Drying areas (yes, it does sometimes rain here!) are usually available and many places are 'pet-friendly'. Do ask if you have any such requirements.
You can search for local accommodation, both
Bed & Breakfast and
Self-Catering, on the Kirkby Stephen website.