Where is knox community hospital located? Knox Community Hospital. 1330 Coshocton Road. Mount Vernon, Ohio 43050. If you need to contact one of our physician offices or central registration to reschedule an appointment or procedure please use the phone listing below as a resource.Why choose knox community hospital mount vernon? Our dedicated staff, along with a strong team of volunteers, is committed to providing personalized, high-quality care. Knox Community Hospital How do i contact kch for further assistance? Looking for a service or office not listed above? Please email us at [email protected] for further assistance.