Step 1: Put the clues together.
This woman had a complicated delivery (her last childbirth) six years ago and never had a medical checkup after that. She now has cold intolerance and hoarseness, pointing to low thyroid hormone, lethargy, pointing to low cortisol, and vaginal atrophy with loss of axillary and pubic hair and vaginal itching, pointing to low estrogen and androgen.
Step 2: Name the condition.
A difficult delivery is often complicated by severe postpartum bleeding, which can starve the pituitary gland of blood and destroy part of it. This is Sheehan syndrome, a form of hypopituitarism that follows postpartum hemorrhage. Because the pituitary makes several trophic hormones, its damage causes a combined shortage of thyroid stimulating hormone, adrenocorticotropic hormone, and the gonadotropins that drive the ovaries, which explains why the thyroid, adrenal, and gonadal systems are all underactive at once.
Step 3: Work out what treatment does and does not help.
Because the hormone shortages are due to gland failure, replacing each missing hormone brings the system back toward normal. Thyroid extract corrects the low thyroid hormone state, easing the cold intolerance and hoarseness. Prednisolone, a corticosteroid, replaces the missing cortisol driven by low ACTH and treats the lethargy and weakness. Estrogen replaces what the ovaries are no longer being told to make, reversing the vaginal atrophy, itching, and loss of body hair.
Step 4: Explain why insulin does not help.
Sheehan syndrome is not diabetes, and it does not cause high blood sugar. In fact, the loss of cortisol and growth hormone, both of which normally push blood sugar up, makes these patients more sensitive to insulin and prone to low blood sugar. Giving insulin would not correct any of her symptoms and could instead push her into dangerous low blood sugar.
Step 5: Final answer.
Insulin is the one agent among the choices that does not improve, and can actually worsen, her condition.
\[ \boxed{\text{Insulin}} \]