。
She just can't manage without milk.
Literal
She [topic-は] milk [without-なしでは] does-not-suffice.
~なしではすまない is a useful pattern: 'cannot get by / be settled without X.' The verb 済む here doesn't mean 'apologize' (the homophonous 済まない / すみません) but rather 'come to a conclusion, suffice, be settled' — its older, broader meaning. Negated, it says the situation can't be resolved without the missing element. The は after なしで adds emphasis or contrast, like saying 'without it, no way.' Japanese has several similar 'cannot do without' patterns: ~なしでは済まない is grumblier and more colloquial than the formal ~なしには成り立たない. Milk consumption rose sharply in postwar Japan thanks to the school lunch program (給食), which has popularized 牛乳 across generations.