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No. You will not recover anything from anybody. Each passenger has 100% responsibility to have all necessary travel documents in their possession. Period. If you pursue your air carrier, you run the risk that someone more senior looks at the situation and figures out that they really ought to pursue you for the cost of getting your wife back to the US. Leave this one alone.

The sole reason for the air carrier checking documents is to prevent the carrier from being fined and otherwise punished for delivering your wife without proper documents. That has nothing to do with you.

As it stands, you were treated exceptionally well. Had your wife been found out by the carrier, she would not have been allowed to fly and many carriers would have cancelled her ticket. In other countries, your wife might have been held in custody until turned over to your air carrier for return and the air carrier might have sought to recover any fines and the cost of a one-way walk-up ticket from your wife.

While this is an unfortunate situation for your daughter who has no capacity to understand what went wrong, let alone do anything about it if she did understand, I cannot stress enough the importance of doing your own document checks and making 100% certain and then checking again that every passenger has every document required.