Rapid developments in the fields of information and communication technology and microelectronics allowed seamless interconnection among various devices letting them to communicate with each other. This technological integration opened up new possibilities in many disciplines including healthcare and well-being. With the aim of reducing healthcare costs and providing improved and reliable services, several healthcare frameworks based on Internet of Healthcare Things (IoHT) have been developed. However, due to the critical and heterogeneous nature of healthcare data, maintaining high quality of service (QoS) -in terms of faster responsiveness and data-specific complex analytics -has always been the main challenge in designing such systems. Addressing these issues, this paper proposes a five-layered heterogeneous mist, fog, and cloud based IoHT framework capable of efficiently handling and routing (near-)real-time as well as offline/batch mode data. Also, by employing software defined networking and link adaptation based load balancing, the framework ensures optimal resource allocation and efficient resource utilization. The results, obtained by simulating the framework, indicate that the designed network via its various components can achieve high QoS, with reduced end-to-end latency and packet drop rate, which is essential for developing next generation e-healthcare systems.