The South Korea international was allowed to leave the Liberty Stadium last season by former manager Michael Laudrup to join the Black Cats on a season-long loan deal.
Ki thrived for Sunderland under Poyet and played an integral role in their survival quest, but was welcomed back to Swansea by Monk at the end of the spell and is now a first-team regular.
Monk told reporters: "There is definitely a lot to thank Gus for. He gave Ki a good role in the team and obviously that gave him confidence to play as well as he did in the difficult situation Sunderland were in.
"He deserves big credit, he highlighted what Ki had and I'm delighted to have him back here having signed a new contract."
The Swans face Sunderland in their Premier League clash at the Stadium of Light later today.