Which king of Babylon sent letters and a gift to Hezekiah after his recovery?

Answer: Merodach-baladan

Merodach-baladan son of Baladan king of Babylon sent envoys because he had heard of Hezekiah's illness and recovery.

“At that time Merodach-baladan son of Baladan king of Babylon sent letters and a gift to Hezekiah, for he had heard about Hezekiah’s illness and recovery.”— Isaiah 39:1 (BSB)