、。
As a result, late in May she was diagnosed with anxiety neurosis and insomnia.
Literal
Its result, May late-period [at-に] anxiety-neurosis [or-や] insomnia [quot-と] was-diagnosed [polite-されました].
その結果 '(as) a result / consequently' is the stock connector in causal narrative. 下旬 '(the) last ten days of a month' — Japanese traditionally divides the month into 上旬 (1st–10th), 中旬 (11th–20th), and 下旬 (21st onward). ~や for non-exhaustive listing. 診断される is the passive of 診断する 'to diagnose' — standard medical reporting usage. The や (rather than と) suggests the doctor may have diagnosed other things too.