having spent the night in the water 是现在分词短语用作状语.根据上下文,既可以理解为时间状语,也可以理解为伴随方式状语；前者说明她向海岸游去的时间,后者补充说明向海岸游去的时候伴随着的行为方式”.
可以作如下改写：
1. 用 and 改写为并列句：
She had spent the night in the water and swam to the shore. 她在海水里已经度过了一个夜晚,接下来向海岸游去.
2. 用 when（= at that time）改写,相当于并列句（从句不可以提到主句前）：
She swam to the shore having,when she had spent the night in the water. 她向海岸游去,此时她在海水里已经度过了一个夜晚.
3.改写为时间状语从句：
After/when she had spent the night in the water,she swam to the shore. 她在海水里已经度过了一个夜晚以后向海岸游去.