The easiest of the questions here to answer is the last question. "How much of the problem should she share with senior managers or should she keep quiet in the hope that she will turn the staffing problem around without anyone being aware that the department was struggling?"
It doesn't seem that anyone, at any level, will really benefit if Ms. Morphine keeps a lid on this worsening situation in her conversations with her superiors. The desire to avoid confrontation, and to avoid being the bearer of bad news, is understandable in human terms but basically inexcusable in a work setting. The efficient working of the pharmacy is in the interests of everyone in this very large organization; it is also evidently an organizational feature that the higher-ups are willing to take for granted. Simply put, neither they nor Ms. Morphine can continue to labor under the perception that "things will get better," because there is little evidence to indicate that they will on their own. Obviously, Ms. Morphine requires some support beyond her own department here, and some of the below recommendations really require a certain amount of support--both budgetary and in terms of personnel--beyond the pharmacy division itself.
The scenario does not identify many reasons for the problems, save that the pharmacy department has a bad reputation. It is possible that this could be combatted directly through better marketing of the department as a place to work, though previously earned reputations change little without a certain amount of evidence that positive change has occurred. A strong recruitment strategy is needed here, and can be summarized in three steps.
Appraise the reasons for high turnover. If there are currently available evaluations of job satisfaction, or if evaluations will have to be newly conducted, a hard look at the data about what employees find particularly frustrating about working in the department is necessary. Without knowledge of why turnover is high, correcting the problems leading to turnover will be impossible. Whatever is behind the high turnover, it is reasonable to assume that if turned-over individuals are proceeding to work in other departments, they are spreading bad vibes (deserved or undeserved) about what it is like to work in the pharmacy division.
Develop a program for filling positions. There are proven ways to attract employees to positions. Perhaps the nature of the work itself, coupled with the recent crisis, justify an increase in wages for new hires that might encourage people to move departments or to choose this pharmacy over a competitor in the case of new graduates. A better training program for agency employees (who I am assuming are equivalent to "temps" in the States) could lead to temporary employees making the jump to full-time, which is good if they are capable, and increase the efficiency of the pharmacy in the meantime. Additionally, a more aggressive recruiting program for new graduates could be undertaken; graduates typically are interested in positions that seek them out, and there are a number of good models for actually recruiting good staff as opposed to ending up with an applicant pool. Whatever the case, this program will likely require a fairly large amount of help from people beyond the pharmacy department; this is acceptable because of the fact that the pharmacy division is essential to the workings of the hospital as a whole.
Redesign responsibilities to make work more satisfying. The pharmacy does have some employees left. Presumably, these individuals will eventually move on if the problems causing high turnover aren't addressed. It is necessary to redesign existing job responsibilities so that work is not as dull, or repetitive, or unfulfilling, or whatever the case may be. Redesigning jobs will increase satisfaction as well as making the available positions more attractive to new hires. Perhaps creative solutions, such as cross-departmental appointments and cross-training, could be used to fill some of the issues and increase the image of the department within the larger organization.
In the end, it's unclear at this point what exactly is causing the turnover and bad reputation of the department. As the management consultant indicates, though, it's not unlikely that a good bit of the issue has to do with Ms. Morphine herself; high turnover and a lack of satisfaction typically originate from some negative management practices. Opening the dialogue with her superiors will certainly increase the possibility for a learning experience here, and will likely eventually lead to some discovery about what could be changed in Ms. Morphine's own management practice.