Question

A rapid response is called for a $70$-year-old woman admitted to the medical ward who has developed worsening respiratory distress. She was admitted two days ago for acute pyelonephritis and has been receiving intravenous piperacillin-tazobactam and vancomycin. Her medical history includes diet-controlled type 2 diabetes mellitus, Hashimoto thyroiditis, and severe chronic obstructive pulmonary disease.

Currently, she is confused and lethargic. Her temperature is $38.4^\circ\text{C}$ ($101.1^\circ\text{F}$), blood pressure is $92/58\text{ mmHg}$, pulse is $118/\text{min}$, and respiratory rate is $28/\text{min}$. Pulse oximetry shows $86\%$ on $4\text{ L/min}$ oxygen via nasal cannula.

Physical examination reveals coarse rhonchi and decreased breath sounds bilaterally, with no jugular venous distention. The point-of-care echocardiogram demonstrates a normal left ventricular ejection fraction ($60\%$) and no significant valvular abnormalities.

Laboratory results are as follows:
Leukocyte count: $16,500/\text{mm}^3$
Serum sodium: $138\text{ mEq/L}$
Serum potassium: $4.1\text{ mEq/L}$
Serum chloride: $102\text{ mEq/L}$
Serum bicarbonate: $14\text{ mEq/L}$
Blood urea nitrogen: $32\text{ mg/dL}$
Serum creatinine: $1.8\text{ mg/dL}$
Thyroid-stimulating hormone (TSH): $0.2\text{ }\mu\text{U/mL}$ (normal: $0.4-4.0$)
Free T4: $0.6\text{ ng/dL}$ (normal: $0.9-1.7$)

Arterial blood gas analysis on room air:
pH: $7.28$
pCO $_2$: $30\text{ mmHg}$
pO $_2$: $54\text{ mmHg}$

Blood cultures drawn on admission are negative, but urine cultures have grown $>100,000\text{ CFU/mL}$ of *Escherichia coli*. The isolate is resistant to ampicillin but susceptible to ceftriaxone, cefepime, and piperacillin-tazobactam.

The patient is urgently intubated for hypoxic respiratory failure. Following stabilization of her airway, which of the following is the most appropriate next step in the pharmacologic management of this patient?