。
She showed me into the room.
Literal
She [topic-は] me [object-を] room [into-に] invited-in.
招き入れる is a compound verb fusing 招く ('to invite, beckon') with 入れる ('to put in, let in') — together 'to invite/usher [someone] inside.' Compound verbs of this type built on 入れる are productive: 引き入れる ('pull in, recruit'), 連れ入れる ('bring in'), 取り入れる ('adopt, incorporate'). に here marks the destination of the inviting-in motion. The grammatical object of 招き入れる is the person being shown in, marked with を.