The 20-year-old joined Rovers on loan at the end of November and, having scored two goals in his first eight appearances, has impressed sufficiently to earn a move.
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"I've enjoyed my time here, we've a great group of lads and I think I've settled in well," he told the club's official website.
"I'm happy to be here and I'm really looking forward to the next two-and-a-half years that I've signed for.
"Everybody has made me so welcome here, I feel comfortable here, I'm happy to make the deal permanent."
Despite featuring regularly for the United reserve team, King made just two appearances for Sir Alex Ferguson's senior squad, both as a substitute.
His previous loan moves included spells with Preston North End, Borussia Monchengladbach and Hull City.