Cap Space
In case you did not know, the Tampa Bay Lightning essentially have zero cap space. Bringing in Kane without any salary retention would carry a $10.5 million cap hit. Even if another team or teams ate 50% of that money, Tampa Bay would still need to clear over $5 million in cap space to make way.
In order to clear out that much room, the Lightning would have to move out somebody significant to their chances of winning the Stanley Cup. That is something that they have been able to stay away from in recent years, so it is very hard to picture them moving on from one of their more important pieces.
The only roster player the Lightning have sent the other way in a trade at the last three trade deadlines has been Mathieu Joseph last season in the trade for Nick Paul. Joseph turned out to be a player the Lightning could live without in order to upgrade and it would take a lot more than moving out somebody like Joseph to make room for Kane.