。
She kept badgering to go home.
Literal
She [topic-は] eagerly want-to-go-home was.
しきりに ('repeatedly, eagerly, intently') is a versatile adverb that intensifies whatever predicate it modifies — here ramping up 帰りたがっていた from a mere 'wanted to go home' to 'kept persistently wanting to go home / was constantly asking to leave.' The desiderative ~たがる ('appears to want to') is reserved for third-person desires, since speakers can't directly access another person's wants — Japanese politeness routes around this with ~たがる, which marks observable behavior rather than direct mental access. Combined with ~ていた (past continuous), the sentence describes her sustained, visible eagerness to leave.