Solder? No, no, no! These are press-on connectors.
Connect black wire to connector closest to outside of NUC. Connect gray wire to other connector.
Make sure wire connector is centered over board connector before pressing. If it's not and you press too hard, you could damage the wire connector.
Hope this helps!
The recommendation given by N.Scott.Pearson was totally correct.
Were you able to connect the wires back into the Wi-Fi card?