。
She slipped inside the door.
Literal
She [topic-は] door [genitive-の] inside [to-に] [entered-and-went-はいっていった].
はいっていった stacks two motion verbs: 入って ('having entered,' te-form of 入る) plus いった (the past of 行く, here functioning as the auxiliary ~ていく for 'do-and-go-away from speaker'). The whole compound captures the trajectory of crossing the threshold and proceeding inward, away from where the narrator stands. ドアの内側 ('the inside of the door') uses 内側 as opposed to 内 alone — 内側 emphasizes the inner-side surface or zone, while 内 is more abstract.