Codex Alexandrinus is a 5th century uncial manuscript, discovered in 1627, 16 years after the publication of the KJV. It contains Mark 16:9-20.
Codex Washingtonensis is a 4th/5th century uncial manuscript. It contains Mark 16:9-20.
So when people tell you that the oldest and best manuscripts (i.e., Vaticanus and Sinaiticus) don't contain Mark 16:9-20, they're overstating their case.
Plus, the Peshitta (Syriac) and other old translations (e.g., Coptic, Georgian, Old Latin) have it.