After removing stuck SIM card, phone doesn't recognize SIM

This has happened before: someone drops a nano SIM card into the open slot (without the tray!) and you have the dubious pleasure of removing it. I did so by opening the iPhone, removing the front cover and (gently) pushing the card from the inside out using Apple's card tray opener.

Unfortunately, the phone didn't recognize the card. Even more unfortunately, it also didn't recognize another SIM card from some other iPhone 5(s).

The folks from the Apple genius bar told the owner they couldn't repair it and just offered an exchange (amounting to 249 Euros).

Now, what could I have possibly damaged and how can I fix it?