United had made a bid of more than £200,000 for the forward, but now have just two days to look elsewhere if they are to strengthen their side ahead of the transfer deadline on Friday.
"Akpo is a big, powerful striker and will give us plenty of extra ammunition in attack this season," Owls boss Brian Laws told the club's official website.
"We have been creating plenty of chances in recent games and I feel that he is the kind of player who can convert those opportunities.
"He is good in the air and his pace and power caused us problems when we played Port Vale in pre-season."
Sodje netted 16 times for Vale last season after moving on a free transfer from League Two side Darlington.