“Find to her cost”

It is an idiomatic expression meaning:

to someone's cost:

With loss or disadvantage to someone.

• ‘without programmes to play on it, the cleverest machine is useless—as some hardware manufacturers already know to their cost’

Also: at someone’s expense

ODO

"...as she learned, to her detriment."

She found out that it was a bad idea, and "to her cost" means that the learning of that lesson cost her something. You can find equivalent phrases using "surprise", "dismay", "consternation", "chagrin", and "detriment". It might be a shortening of "...leading to..."?