Killing somebody (more likely to be described as something like "missing opportunities that may have contributed towards the survival of a critically ill patient") is unlikely to be seriously investigated unless a lot of people are screaming loudly enough about it. Even if their manager comes out and says that they are dangerously incompetent then what would actually happen is that they would get put on an improvement plan instead of being fired.
This means that the staff member gets somebody else from the department trailing them around until they either don't make any mistakes for long enough or make massive and (provably) lethal mistakes in front of several people.