Wenger made a loan move for the Sweden international after seeing injury deplete his midfield ranks at Arsenal, with Aaron Ramsey and Jack Wilshere sidelined.
But Kallstrom, who has made 108 appearances for his country, arrived from Spartak Moscow with a back injury and faces a month out.
Defending his decision, Wenger told Sky Sports 1: "The earliest he could be available would be the end of February and the latest would be mid-March."
"We have to establish the real truth about this story. He arrived here and fell on his back two days before.
"First of all it's very difficult to identify a player, who can come in, of that stature and then we made a medical (check) of his back and we found out he had a little micro-fracture.
"We have all played with micro-fractures.
"Then Spartak Moscow accepted to share the costs - as long as he's injured they will pay - and we had a decision to make."
Asked if he considering pulling out of the deal, the Arsenal manager said: "Yes of course it crossed my mind. I would not have signed him if we had two or three more days to do something, but it was Friday night at five o'clock, so it was [a case of] you [sign] nobody or you do it under these conditions.
"I decided to do it because we might, because of the number of games we have now in February, need the players in March or April.
"There is a possibility [that he may not play], but as well there is the possibility that he scores us the winning goal that might be vitally important."
Wenger added: "We could have gone without, but first of all you must identify a player who has the quality to play for us in January.
"Then, because we have the number of players needed in that area, [they have to be] not locked in a deal of three or four years, where you pay the players to do nothing.
"So to find a player on a free loan, of that quality, on Wednesday morning until Friday night, is not easy."
Nevertheless, Wenger accepted it was a move which could come back to haunt Arsenal as they go in search of a first trophy since 2005.
"At some stage in our job you have to make a decision. Are you wrong or right? You will only know at the end of the season," he said.
"I made this decision because, since the start of the season, all our central midfielders have missed a part of the season and it can happen that a few of them miss the next part [of the season] together."
Arsenal had been linked with Schalke's £37million-rated German midfielder Julian Draxler, but a big-money bid never materialised.
Wenger said: "We are not against spending, but at the moment we are focused on the second part of the season, and at the end of the season we will see what we will do.
"You have announced a lot of transfers in your papers that have not happened, in many places, that means it is not as easy as it looks, even with money, to buy the right players in that period."