非谓语：attracted by the beauty of the city,she decided to stay

Attracted by the beauty of the city
=Being attracted by...
=Having been attracted by...
=As she was attracted by...
意思都一样,但第一种最简练、地道.你只需要熟悉并背下这个很有用的句型就行了.
文中既然说“another days”,即暗示她已待了两天,2+2=4.
你举的第二个句子是错的.可改为：
Looking into the case,...
或：As he looked into the case,...
同理,第一种写法更好.