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Strip about 1/2" of insulation from both wires.
Twist the wires together and insert them into the twist-on wire connector. Turn the connector several times until the cap feels secure, and you can tug on it without it falling off.
After testing the lights and making sure the bad socket fixed the problem, consider putting in some silicone sealant (or museum wax) into the cap in order to keep moisture out and prevent the wires from corroding.