很高兴为您解答!
【译文】
1、当我走进这间屋后,我的眼睛迅速适应了昏暗的房间.可否译成：My eyes quickly adapted to the darkness shortly after entering this room.或My eyes were quickly accustomed to the
quickly与shortly重复了,留一个即可.其他没有什么问题
2、他终于承认这是他的错.可否译成：He finally acknowledges that it was he fault.
谓语动词acknowledge最好用一般过去时态,另外it was his fault.
简单点翻译：He finially acknowledged his fault.