Hi,
Even I am in the healthcare sector, and in our hospital, we have a proper HR Department established. You can start with proper personal file updation, a savior application installed for time office work, and payroll work (PF, Mediclaim, and various other things). Various employee benefits programs, proper policies for travel, recruitment, appraisal, compensation, T&D, internship (because in hospitals we charge interns in some fields and give stipend in some, so it's a must), allowance policies like for day manager, night manager, night allowance, risk allowance policy (for employees working where radiations are emitted like X-ray and all, we give special allowance), and many more. You just have to start, and you'll find no end.
If these policies are there, and you can start working on getting some recognition for your hospital like NABH, JCAI, and many more.