I assume you're not asking how to proceed in an actual lawsuit or prosecution, and are just posing this as an information question about law. In this scenario, Jack wants to sue as many people as possible, and to have as many people as possible prosecuted.
Jack can sue the involved hospitals and doctors for negligence. The basis of the suit would be that the defendants failed to exhibit the required degree of professional care under the circumstances, resulting in Jill's death. Crucial to the argument is that they did something which they clearly should not have done that contributed to her death, or failed to do something that they clearly should have done. Assume that the infectious agent was stachybotrys ebola (currently non-existent), that 40% of houses in the county have this agent, and that it can be cured with a two-week course of tetracyclin. Patients generally present certain symptoms when they have the condition, and there is a simple test that confirms the diagnosis. In this instance, the patient did present those symptoms and didn't present conflicting symptoms.
The legally-relevant question is, what would the average physician customarily do under these circumstances. Since this is a "mysterious" agent, they must have failed to diagnose the condition. Again, we have to be specific about what the doctor(s) did wrong, because failing to diagnose is not per se negligence. If the standard strong indicators of the infection is bleeding gums (which the patient did report) and lowered BP, and the doctor didn't look or take BP, that is probably not up to the standard of care that would customarily be given by the average physician in that situation (i.e. walk-in clinic doctor, not fictitious TV super-sleuth doctor).
Jack doesn't have to be positive about the doctors never mentioning such a test, because it's all in the record. Maybe after the fact somebody mentions this infectious agent and he hires a medical sleuth to determine that the agent is present in the house. He gets the idea to sue everybody and subpoenas records right and left, and discovers that (a) although they didn't identify that agent, all of the objective facts clearly indicate that was the cause of the ailment and the cause of death and (b) they had the evidence required to correctly diagnose the situation, but did not use a high enough standard of professional care.
The described scenario is rather far-fetched, because it appears based on the unanimous decisions of 6 independent doctors and 2 hospitals that they all were in fact using the customary standard of care in the profession.
The legal outcome will depend in part on the state where this took place. There is a "two schools of thought" doctrine in some states which allows that there might be competing medical theories, so even though 60% of physicians would do X, 40% would do Y, and doing Y is not a negligent action (those numbers aren't part of the legal doctrine, they are there to clarify the kind of situation where this could apply).
Jack probably doesn't do anything other than make some phone calls and write some checks. It is extremely unlikely that he would "discover" this agent in the house, and more likely he would just call an attorney in the hope of getting something from the doctors. The attorney (experienced in such matters) hires medical investigators that lead to the factual allegations which are the foundation of the lawsuit.
If the doctors were really that bad, then they could be sued for millions of dollars, and possibly have their licenses to practice stripped, depending on how shocking the neglect was.