I don't think he will, at least until the whole White Walkers thing has been handled.

And Dany is wrong, he is not acting out of pride. He is just doing a piss poor job of explaining himself to her. Jon doesn't need a Hand, he needs a communications director.

If he'd bent the knee now, he would be obliged to do her bidding (or be considered a traitor if he refused), which would involve actively participating in her war against the Lannisters. His troops in the North are already spread too thin as it is, and he can't afford to send even one of them south.

He will definitely need to find a way soon to convince Dany and the rest of Westeros that the White Walker threat is real, though (as if the birth of Dany's dragons and all the resurging magic didn't give the people enough hints that something is very, very wrong ). The only living eyewitnesses are Jon, Sam, the survivors of Hardhome, Bran and Meera. The rest of Westeros probably considers all of that "fake news".

Edit:
If GoT were a romcom, Dany would bend the knee to Jon in the end, proposing to him.