I'm calling yes, she played him.
Before going to Kings Landing Sansa never had any time for her bastard brother, taking her lead on that from her mother.
She was genuinely glad to see Jon, but then he was her only hope of escaping Ramsay.
The only way she and Jon have of escaping Ramsay is to kill him first. When it was apparent they didn't have the forces she PM'd Little Finger.
Why didn't she tell Jon about Little Finger?
She didn't believe in Jon enough to think he could win the battle, even with the force from the Vale. She couldn't afford a winter siege, and Jons un-augmented force was enough to pull Ramsay out into a toe-to-toe battle.
She believed Ramsay would gain the upper hand, and knowing he was completely without mercy, guessed he would encircle Jons forces and finish them.
So, the Vale arrives in time to rout Ramsays forces, attacking unprotected flanks, Ramsay flees, but is quickly captured and killed.
So obviously Sansas primary goal is achieved, Ramsay is dead.
It's the secondary outcomes that are most interesting, as they leave Sansa in an excellent position.
- She is the legitimate warden of the north.
- She can "forgive" her vassal lords that declared for Ramsay, but
demand tribute of money and men for the favour, quickly building her
resources to a point where she's somewhat safe.
- The wildling army on her doorstep has been decimated.
- The only force of opposition left near the north are the Freys,
and they're far enough away, and committed to the south, that
they're unlikely to be an immediate problem.
The fifth beneficial secondary outcome, that could easily have happened but didn't, would have been Jons death in battle.
With Jon gone Sansa is the eldest (she believes only) living child of Ned Stark. As a member of the Nights Watch Jon wasn't a threat, but now that his watch has ended, the North is as likely to legitimise Neds bastard and declare for him as they are for "a slip of a girl".
It'll be interesting to see what happens there, does she actively want him dead? Or just out of the way?
Did she think all those things through, I don't know, but she knew she'd be better off not giving Jon the forces from the Vale.
It's been a long time since she's been the empty headed girl that left Winterfell. She's learned a lot dealing with Joffery, Tyrion, Little Finger and not least Ramsay.