she was so angry at all that he was doing（ that）（我写成what）she

walked out，and closed the door
（如果walk out改成put up with）是不是前面的that就是so that 的that，就可以写what了？

不改，这句句子是so……that的结构
all先行词，that后面连接了一个定语从句：he was doing，that是从句的宾语。
最主干的是She was so angry that she walked out
后面的that就是你改成what的其实是so……that结构中的that。

不能改成put up with，首先是句意不通，其次so一般不能孤立使用，要么so……that，so……as。你那样改是定语从句里面再套一个定语从句么？很奇怪。so后面就只有一个angry了，不能这样说的。