If we can figure out how old Jamie is in Court Martial, that should help pinpoint Kirk's career timing a bit. She would have been named somewhere between meeting Ben at the Academy and Kirk's having pissed him off on Republic.
Short Answer: I always assumed Jamie was fourteen but can't find any statement to that effect.
Long Answer: Numerous calculations fail to narrow down Jamie's possible age in "Court Martial" very narrowly.
In "Court Martial":
STONE: Let us begin with your relationship with Commander Finney. You knew him for a long time, didn't you?
KIRK: Yes. He was an instructor at the Academy when I was a midshipman, but that didn't stand in the way of our beginning a close friendship. His daughter Jamie, who was here last night, was named after me.
STONE: It's common knowledge that something happened to your friendship.
So Jamie was named, and thus presumably born, after Kirk became a midshipman at Starfleet Academy.
In "Shore Leave" Kirk meets Ruth and Finnegan who Kirk knew when he was at Starfleet Academy:
KIRK: What's the matter, Bones, you getting a persecution complex?
MCCOY: Well, yeah, I'm beginning to feel a little bit picked on, if that's what you mean.
KIRK: I know the feeling very well. I had it at the Academy. An upper classman there. One practical joke after another, and always on me. My own personal devil. A guy by the name of Finnegan.
MCCOY: And you being the very serious young
KIRK: Serious? I'll make a confession, Bones. I was absolutely grim, which delighted Finnegan no end. He's the kind of guy to put a bowl of cold soup in your bed or a bucket of water propped on a half-open door. You never knew where he'd strike next. More tracks. Looks like your rabbit came from over there.
KIRK: Ruth? Ruth.
RUTH: Jim, darling, it is me. It is Ruth.
(She kisses him)
Captain's log. Stardate 3025.8. Investigation of this increasingly unusual planet continues, and we are seeing things that cannot possibly exist, yet they are undeniably real.
[Rocky outcrop]
KIRK: (into communicator) McCoy, do you read me? Ruth. Ruth, how can it be you? How could you possibly be here? You haven't aged. It's been fifteen years.
RUTH: It doesn't matter. None of that matters.
KIRK: First there was McCoy's Alice in Wonderland where there was supposedly no animal life. And Sulu's gun where there's no refined metal detected. Rodriguez' birds. And then my, well, the two people I saw.
[Bridge]
SPOCK: Any chance these could be hallucinations?
[Rocky outcrop]
KIRK: One hallucination flattened me with a clout on the jaw.
SPOCK: There is one slight possibility, very slim, but nevertheless. Captain, what were your thoughts just before you encountered the people you described?
KIRK: I was, I was thinking about the Academy. My days
FINNEGAN: Hey, Jim baby! I see you brought out reinforcements. Ha! Well, I'm waiting for you, Jimmy boy.
FINNEGAN: Get up. Get up. Get up. Always fight fair, don't you? True officer and gentleman, you. You stupid underclassman. I've got the edge. I'm still twenty years old. Look at you. You're an old man.
FINNEGAN: I never answer questions from plebes, Jimmy boy.
KIRK: I'm not a plebe. This is today, fifteen years later. What are you doing here?
This implies that Kirk was a plebe (first year cadet) at Starfleet Academy 15 years before "Shore Leave". It also implies that Kirk met Ben Finney and became his friend no more than 15 years before "Shore Leave" and Finney named his daughter Jamie no more than 15 years before "Shore Leave". So it seems very probable that Jamie Finney was no more than 15 years old in "Shore Leave" and possibly years younger.
But the friendship between Kirk and Ben Finney ended when Kirk was an ensign on the
Republic.
"Court Martial":
STONE: It's common knowledge that something happened to your friendship.
KIRK: It's no secret. We were assigned to the same ship some years later. I relieved him on watch once and found a circuit open to the atomic matter piles that should've been closed. Another five minutes, it could have blown up the ship.
COMPUTER: Ship nomenclature. Specify.
KIRK: United Starship Republic, number 1371.
STONE: Continue.
KIRK: I closed the switch and logged the incident. He drew a reprimand and was sent to the bottom of the promotion list.
STONE: And he blamed you for that?
And:
SHAW: With reference to Records Officer Finney, was there in his service record a reported disciplinary action for failure to close a circuit?
ENSIGN: Yes, ma'am.
SHAW: Was the charge in that instance based upon a log entry by the officer who relieved him?
ENSIGN: Yes, ma'am.
SHAW: And who was that officer?
ENSIGN: Ensign James T. Kirk.
SHAW: Louder, please, for the court.
ENSIGN: Ensign James T. Kirk.
SHAW: Now the Captain Kirk who sits in this courtroom?
ENSIGN: Yes, ma'am.
So Kirk had the rank of ensign when he reported Finney, ending their friendship. According to my rough calculations, Kirk would normally have been graduated from the Academy and commissioned an ensign about 3.75 years after Kirk entered the Academy.
So if Kirk entered the Academy about 15.00 to 15.99 years before "Shore Leave" he would have graduated and become commissioned an ensign - if after the normal period - about 3.75 years later and thus about 11.25 to 12.24 years before "Shore Leave". Since all TOS episodes happen during a five year mission which should be less than 5.99 years long, no TOS episode should happen more than 21.98 years after Kirk entered the Academy.
The latest possible time for Ben Finney to name Jamie after Kirk was the time when Kirk reported Finney and ended there friendship, and the latest possible time for Ensign Kirk to report Finney was the day when Ensign Kirk was promoted to lieutenant.
According to "A Private Little War":
SPOCK: Aside from that, you say it's a Garden of Eden?
KIRK: Or so it seemed to the brash young Lieutenant Kirk on his first planet survey.
SPOCK: Class M in all respects. Quite Earth-like.
and:
KIRK: When I left there thirteen years ago, those villagers had barely learned to forge iron. Spock was shot with a flintlock. How many centuries between those two developments?
So Kirk was a lieutenant making his first planetary survey "thirteen years" before "A Private Little War", which might be 13.00 to 13.99 years, or maybe 12.50 to 13.50 years, or maybe 12.0 to 14.99 years, depending on how Kirk thinks about and reports time intervals.
So Kirk was already a lieutenant making a planetary survey 12.00 to 14.99 years before "A Private Little War", and he was promoted from ensign to lieutenant an unknown period of time before making that planetary survey, and Jamie Finney must have been born and named after Kirk an unknown period of time before Kirk was promoted from ensign to lieutenant.
Therefore Jamie Finney must have been at least 12.00 years, and possibly almost 20 years, old in "A Private Little War".
If James Kirk graduated from the Academy and was commissioned an ensign about 11.25 to 12.24 years before "Shore Leave", and if it took Kirk about 0.01 to 4.0 years to be promoted to lieutenant, he was promoted to lieutenant about 7.25 to 12.23 years before "Shore Leave", and that was the very latest that Jame Finney could have been named after James T. Kirk. Therefore Jamie Finney was about 7.25 to 15.99 years old in "Shore Leave" and at least 12.00 years old in "A Private Little War", which thus must have been 0.01 to at least 4.75 years after "Shore Leave".
If Kirk was promoted from ensign to lieutenant about 7.25 to 12.23 years before "Shore Leave", and if Kirk was a lieutenant 0.01 to 4.00 years before making the planetary survey, Kirk made the planetary survey about 3.25 to 12.22 years before "Shore Leave". If lieutenant Kirk made the planetary survey about 12.00 to 14.99 years before "A Private Little War", "A Private Little War" should be about 0.01 to 11.74 years after "Shore Leave". Therefore Jamie Finney should be aged about 7.26 to 27.73 years old in "A Private Little War".
But of course two different episodes in the five year mission should be no more than 5.99 years apart at the very most. So if "A Private little War" is after "Shore Leave", Jamie can be no more than 13.24 to 18.22 years old in "A Private little War", and possibly years younger than that. And if Jamie must be 12.00 to 18.22 years old in "A Private little War", she can be no younger than 6.01 to 12.23 in "Shore Leave".
If Kirk graduated from Starfleet Academy and was commissioned an ensign after about 3.75 years, and was an ensign for at least 0.01 years before being commissioned a lieutenant, and was a lieutenant for at least 0.01 years before making his planetary survey, and if the planetary survey was at least 12.00 years before "A Private Little War", "A Private Little War" must have been at least 15.77 years after Kirk Entered Starfleet academy.
Since "Shore Leave" should be 15.00 to 15.99 years after Kirk Entered Starfleet academy, "A Private Little War", when Jamie should be 12.00 to 18.22, should be sometime between 0.22 years before "Shore Leave" to 5.99 years after "Shore Leave:, thus making Jamie aged 6.01 to to 18.44 in "Shore Leave. Since Jamie Finney should be about 7.25 to 15.99 years old in "Shore Leave", she should be 7.03 to 21.98 in "A Private Little War".
Most
Star Trek fans would probably imagine that "Court Martial" happens within one year of "Shore Leave", and thus when Jamie Finney is aged about 6.25 to 16.99 years old. And in "Court Martial" she looks closer to the older end of that age range than the younger end.
I assumed that Jamie was fourteen in "Court Martial" but I have not found any statement of her age to back that up.