(All opinions expressed here are based upon my one-time viewing of Star Trek: Into Darkness at the cinema. I cannot remember the exact dialogue)
At the beginning of the movie, when Kirk asks Bones what Spock would do in the same situation, Bones replies "he'd let you die".
At this point, Kirk should have said "that's why I need him". Because Kirk chooses to save Spock, the audience is allowed to believe that he respects Spock's decision-making and values the opposing thought process even more than his own life. By adding "that's why I need him", we can get a larger dramatic payoff from a later scene, which will strengthen the idea of their core relationship.
I would have cut the part where Spock asks Spock Prime (Leonard Nimoy) for help. I've got no time for senseless crowd pleasing. The scene added nothing.
Instead, Khan should present the terms "hand over the soldiers or the Enterprise and all its crew are destroyed". Spock will consider this, look around to his despondent crewmates, and tell Khan that he cannot hand over the soldiers.
Kirk, while still aboard the opposing ship, will smile as he realises Khan's bluff has been called. Khan needs the soldiers, so he can't destroy the Enterprise where they reside. The Enterprise crew will relax.
Khan will then propose a counter-offer: send the soldiers, or he will kill Kirk. Spock has ten seconds to reply, and as Khan puts the gun to Kirk's head and begins to count down, Kirk will tell Spock that it's okay, that he knew this moment would arrive, some day, and he needed Spock to be in captains chair to ensure the rest of the crew's safety (thus fulfilling the "that's why I need" him presupposition from the beginning of the movie).
To Kirk's outrage, Spock will cave. He will tell Khan to return Kirk in exchange for the pods.
Kirk will arrive on the Enterprise and run at Spock, shouting that he gave him the captains chair to make decisions that weren't based soley on his heart, to make the most logical decision. And he'll ask "how could you throw it all away, just to save me?" and Spock will reply "I didn't."
The opposing ship will blow up and we find out about the plot to remove the soldiers and send the missiles armed.
In this version of events, we get a moment to feel that Spock was trying to save Kirk, which, hopefully, as human beings, the audience should empathise with. But the twist comes from Spock admitting to Kirk that he didn't act purely to save him - it was incidental. It's a cold sentiment but one that Kirk values. His decision to reinstate Spock was the right one, and the idea that Kirk needed to save Spock at the beginning of the movie is fulfilled. Spock chose the logical answer that did the greatest good, it's just by chance that Kirk was saved to accomplish that. Had another route have created the most good but lost Kirk's life, that route would have been chosen instead.
I'd want this idea to be reiterated at the end of the movie also. Maybe at that inauguration/ceremony/funeral thing (whatever that was) or back on the Enterprise. Kirk can say something like "you know, you could have just sent the warheads over at the first time of asking, before he threatened my life..." (so he addresses that potential plot hole of not sending the armed warheads on Khan's first time of asking) But then he'll add, "ahh but then Kirk here would have bitten the dust aboard the ship with Khan... and you just couldn't let that happen could ya?" (But something less terrible!) He'd be teasing Spock, to Spock's humourless expression. Then Spock will turn solemn and say "one day, a decision I make may end your life, Kirk", and Kirk will say "on that day, it will be the right decision".
This way more weight is added to the characters' relationship. Spock and Kirk are mutually inclusive. Two sides of the same coin. We see a thought process from Spock that means Kirk is not the priority, the greater good is. This is tinged with sadness, the thought of Spock being so brutal about Kirk's death and Kirk's apparent satisfaction with that notion. But that's why it's dramatic.
In the real film, when Kirk is dying and he congratulates Spock on his decision to send the armed warheads onto the other ship, the part where Spock says "it's what you would have done" falls completely flat. There is no previous suggestion of this line of thinking. It does not offer an emotional pay-off without convincing the audience that he has a reason to believe it. Why is that what Kirk would have done? Why does Spock believe he should act in the way in which Kirk would? A few additional lines of dialogue could have completely rectified this and made the film that much more enjoyable.
I'd also remove the unnecessary display of Alice Eve, semi-naked. Kirk can still be a player without shoving it in the audience's face.