I'm not sure you could. TOS didn't have any kind of framing story or overall arc to finish off like DS9 or Voyager. Best you could hope for would be a TNG style ending where you basically go for an "episode Plus", then end it on a "and they went on to new adventures" note.
If I was going to do that... I'd have Kirk bringing the Enterprise into dock for a maintenance, only to find that Starfleet's Commander in Chief is there to tell him he's being promoted to Rear Admiral and given a desk job on Earth. Kirk is bittersweet about it, as it's made clear that his choices are to move on up or retire from Starfleet. He wonders if he shouldn't take the job and turn the ship over to some hot up and coming officer... maybe even take some of his officers along with him to his new job, which would mean promotions for them.
Then there's a surprise attack on the station. The ship is damaged and forced to retreat, the station - and the C in C - is captured. With his warp drive out Kirk has to dodge the enemy which is hunting for him throughout the system, eventually recapturing the station and driving the enemy off in a truly brilliant series of maneuvers. The grateful C in C agrees to give Kirk with his choice of any job in the fleet, and the series ends with his officers wondering what he might do before Kirk enters the Enterprise bridge and says something cheesy like "there's no place like home" as he takes his seat and orders the ship back into the unknown.
Give a man a fire, and you keep him warm for a day. SET a man on fire, and you will keep him warm for the rest of his life...