、。
Just when I thought she'd flatly turn me down, she agreed.
Literal
Lightly will-be-refused [quotative-と] thought-[when-きや], she [topic-は] consent-did-for-me.
思いきや is a classical/literary pattern — the ~きや ending is an archaic inflection meaning 'just when one thought' and always signals a contrast with what actually happened. The phrase survives in modern writing as a set expression, and learners should treat it as an idiomatic unit rather than trying to parse the old grammar. ~てくれる on 承諾する is the benefactive, making clear the agreement was done for the speaker's benefit. あっさり is a state mimetic for doing something lightly, without resistance or elaboration.