The first thing that will happen for sure is that Jack will lose his shirt. This always happens. Jack's sword will work on Inuyasha but won't work on him if Inuyasha steals it. Although the Tetsaiga won't transform into its powerful form, it can still be used to block Jack's attacks since it is still indestructible. I am assuming that they are equal in agility. I see them jumping back and forth at each other, Jack using his sword and Inuyasha using his claws. This is still an even matchup. Even if Inuyasha gets hurt, he has the ability to use that in a move "blades of blood". Jack's sword will protect him though. So basically they are evenly matched and this fight goes on for hours, like most of Jack's fights. I think Jack is the more skilled of the two and therefore he wins. He has more training and experience. He can outsmart Inuyasha.
Update: I just finished watching the second episode of the new set of episodes on YTV. This episode was the one where Inuyasha kills the demon that killed his father. In this episode we learn that because of the changes that Totosai made to the Tetsaiga, Inuyasha can now use the windscar any time he wants to and that the Tetsaiga has an ultimate attack called the Backlash Wave. Inuyasha is able to rapidly fire off windscar attacks. I originally thought that Inuyasha couldn't use the windscar because the explanation they gave. The windscar is formed when Inuyasha's "power field" rubs up against the "power field" of his opponent. I say "power field" because I don't remember what they called it. I don't think Inuyasha can use the Backlash Wave since the explanation they gave said that it uses the demon energy of his opponent. Jack has no such energy. Now that Inuyasha can just fire off the windscar, the sort of stalemate that I mentioned above turns into Jack trying to dodge the attacks. I know Jack has taken on guys that can fire energy blasts before but he'll get tired fast. I don't remember Inuyasha getting tired. Jack wouldn't be able to get close enough to Inuyasha o land any attacks. Another update is that if Inuyasha loses the Tetsaiga, his demon blood will take over and turn him into a ravenous beserker. So even if Jack gets in there and disarms Inuyasha, he will still have to deal with a stronger, fiercer version of Inuyasha. Unfortunately YTV didn't seem to get the new seasons of Samurai Jack so I have no updates for him. So for now Inuyasha is the winner.
Winner: Inuyasha