Jane doesn't go straight to violence. House would. Especially when trapped in an elevator.
I'm not 100% convinced that it would come down to a physical confrontation, but if it does, House definitely wins. He's the one with the weapon, upper body strength, scientific knowledge of precisely where to hit. If he's willing to use the cane against a friend who thwarts his will, think what he would do when faced with another man with some of the same superiority complex, who is as willing as House is to irritate the hell out of people by laying open their hidden motivations. Yeah, House loves to do that himself, but he has zero desire to be flayed in the same way. House's need for dominance is more overt than Jane's. And Jane's survival in a small enclosed space for any length of time with no one but House for company depends mostly on his recognizing that he could provoke House too far before it is too late, and also having his sense of self-preservation operational at the moment he makes the choice.