how to understand?
•••"We've fallen on our feet and no mistake," said Peter.•••

As soon as they had said good night to the Professor and gone upstairs on the first night, the boys came into the girls' room and they all talked it over.
"We've fallen on our feet and no mistake," said Peter.

Question

how to understand?
•••"We've fallen on our feet and no mistake," said Peter.•••

As soon as they had said good night to the Professor and gone upstairs on the first night, the boys came into the girls' room and they all talked it over.
"We've fallen on our feet and no mistake," said Peter.

Answer

It means to be in a good or even better situation after a difficult experience or event.  Another term for it:  "land on your feet."  For example, one might say to a friend:  "I know it was tough losing your job but, give it time, you'll "fall on your feet."  As James pointed out "no mistake" is like saying "certainly" or "no doubt about it."

Answer

"Fall on one's feet" = lucky (as opposed to on your head or stomach, I suppose).
"[make] no mistake [about it]" = certainly