The 20-year-old initially arrived at Ewood Park on loan at the end of November and has scored twice in his eight appearances so far.
The Norwegian international has signed for an undisclosed fee and he has agreed a contract which runs until the summer of 2015, with an option for a further year.
King said: "I've enjoyed my time here. We've a great group of lads and I think I've settled in well.
"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 and I'm happy to make the deal permanent."
King can play out wide or as a centre-forward and has long been regarded as a bright prospect at Manchester United, for whom he made his debut as a substitute in a 1-0 League Cup win over Wolves in September 2009.
However, he has been unable to break into the United side again and spent time on loan at Preston, Borussia Monchengladbach and Hull before arriving at Ewood Park.