。
It seems she had gone many nights without sleep.
Literal
Many-nights sleepless nights [object-を] spent [seems-ようです].
A literary/narrative observation. 幾夜 (いくよ) is a slightly poetic/formal word for 'many nights' — 幾 is an old interrogative-based prefix meaning 'how many' or 'many,' surviving today in set expressions like 幾つ, 幾人. The も attached turns 幾夜 into an exclamatory 'so many!' Note the near-repetition: 幾夜も...夜 — 'many nights... of nights,' redundant in English but perfectly natural in literary Japanese for emphasis. 眠れない夜 ('unsleepable night,' 'a night that can't be slept') uses the potential 眠れる in negative. ~ようです is the speaker's guess from evidence — 'it appears,' 'it seems,' softer and less confident than らしい.