The Internet of Things (IoT) paradigm is drastically changing our world by making everyday objects an integral part of the Internet. This transformation is increasingly being adopted in the healthcare sector, where Smart Hospitals are now relying on IoT technologies to track staff, patients, devices, and equipment, both within a hospital and beyond. This paradigm opens the door to new innovations for creating novel types of interactions among objects, services, and people in smarter ways to enhance the quality of patient services and the efficient utilisation of resources. However, the realisation of real-time IoT applications in health care and, ultimately, the development of Smart Hospitals are constrained by their current Cloud-based computing environment. Edge computing emerged as a new computing model that harnesses Edge-based resources alongside Clouds for real-time IoT applications. It helps to capitalise on the potential economic impact of the IoT paradigm of $11 trillion per year, with a trillion IoT devices deployed by 2025 to sense, manage, and monitor the hospital systems in real time. This vision paper proposes new algorithms and software systems to tackle important challenges in Edge computing-enabled Smart Hospitals, including how to manage and execute diverse real-time IoT applications and how to meet their diverse and strict Quality of Service (QoS) requirements in hospital settings. The vision we outline can help tackle timely challenges that hospitals increasingly face.