Yes
There is nothing the GM cannot do with their NPCs which players can do with their PCs, as far as gameplay actions go, so, declaring details in the middle of the action isn't off the table. And there's no reason the GM wouldn't spend the Fate point for it.
To quote Declare a Story Detail on FC p. 13: "To do this, you’ll spend a fate point."
However, since the GM is the referee of the acceptability of detail declarations, and story detail declarations "work to your character's advantage" by definition (FC p. 13), it begs the question of whether this is a good idea or not. I, as a GM, would make very sure to pay attention to the explicit reference to "player buy-in" which is also spelled out on page 13, because holding the veto power over my own NPC's play options can easily create perceptions of unfairness.
It's not a big thing. It's just a Fate point. As a GM, I don't spend a lot of NPC Fate points anyway, myself. Since I get them every scene, I can already get way more than the players do just by changing scenes a lot. So it pays to be fair here.
To address finer points:
"When would you or wouldn't you involve fate points in the GM's duty of narration? Bearing in mind that the GM is continuously narrating story detail."
"Declaring a story detail" as given on Fate Core page 13 isn't the "duty of narration". It isn't plain-vanilla roleplaying, it isn't "what does my (non-player) character do now", it isn't even world-building. It's spending a Fate point to make the narration go in a direction which works to the advantage of the character you're playing. So, the GM might want to do it for the same reasons a player would want to: The scene isn't going my way, so, I want to declare something like "Of course Jen has something she can pick handcuffs open with." The GM would not do it just to say that there is opposition in the scene, or that the Scar Triad is in town, or that there is a flood which has made the Crab River Bridge un-crossable. The rule is there in order to conjure up a fact of immediate benefit to the acting character in real-time in gameplay.
(Aside: After giving her the hairpin lockpicking tool, I sure hope that you had Jen make a Burglary roll instead of just breaking her out automatically. Spending a Fate point shouldn't get you everything.)
As a GM, I would make sure to be seen spending the Fate point when I invent some bit of narrative which I had not prepared ahead of time and is clearly a mid-scene, mid-action brainstorm to make things easier for my character. A GM can do all kinds of things which surprise their players without spending Fate points, but, when they see us making up a page-13 type New Story Detail specifically for the sake of making the immediate moment in the scene harder for them, it's classy to play by the same rule and pay the point.
"Any of the facts the GM narrates can become an aspect at no cost just because someone thinks it would make sense to turn that fact into an aspect[1] -- naturally the GM couldn't be expected to pay a fate point for all the details they narrate." ([1]: Fate Core p78, Creating and Discovering New Aspects In Play, last paragraph. Or on Fate SRD: https://fate-srd.com/fate-core/using-aspects-roleplaying#creating-and-discovering-new-aspects-in-play)
That's different from the "Declare a Story Detail" rule. First, the details Declared under the Declare rule aren't Aspects, at least not according to the bare rule. As page 74 points out, an important enough fact can be agreed to be made an Aspect, but, that's separate from the Declare rule, and it could be done after the Declaration has been done, but it's still separate.
Second, most of "the details the GM narrates" are not done for the sake of giving their NPCs quick advantages and boons, and that's true of most of the details the players narrate, too. These strictly-narrative, non-big-D-Declare narrative details are there for completely different reasons. They're intended to be the description of the situations the characters find themselves in, and the reactive actions of the characters as they interact with opposing characters who frustrate their goals and efforts. GM's don't spend Fate points just to GM, as has been pointed out, just as players don't spend Fate points to declare their own characters' every action and history detail either - even the ones they make up on the spot. We pay Fate points for those Declarations which work to our character's advantage immediately in a situation where we are seeking to introduce something narratively, something new, something not previously established, something helpful, in the face of pressing challenge.
"So: when does a GM pay fate points for creating story details (if ever)? When do they simply narrate? Certainly there's a difference."
What I personally recommend is "never", for reasons stated above. But certainly there's a difference. To see what it is, let's imagine that "Declare a Story Detail" were not a Fate Core rule and were not one of the ways you can normally spend Fate points (Fate Core pages 13, 80).
In this case, the only way to do something like declaring that "I have just the right tool for the job, natch" or "Naturally I know someone in this wretched hive of scum and villainy we've just entered" is to have a Stunt powering that. Such a Stunt would work the same way whether it's on a player character or a non-player character. With the Declare rule, such Stunts are written to not cost a Fate point. That's the Stunt's very benefit - you can do it without spending a Fate point. In the non-Declare world, it's extremely easy to imagine Stunts like that being written to allow the exact same benefit, only, maybe it would cost a Fate point to power the Stunt, since it would be a benefit which there would be no other way to get. (That's one of the uses of Stunts: for "breaking rules" or creating exceptions to rules. FC page 91.) Again, such a Stunt could be either on a PC or on a NPC and would work the same (cost the same cost, yield the same benefit) either way.
That's the difference between the routine "duty of narration" versus the Declare a Story Detail rule. It's like the difference between narration and rolling the dice. One is what you do to tell the story, the other is a game mechanic you can use to determine whether the story can go a certain way or not. Once you offer the Fate point and negotiate with the table and the table agrees that it's fair and legal (exercise the mechanic), then you go ahead and add your new detail to the living story, and play on from there.
Page 13 says that the GM is the referee, but, when it's the GM who wants to benefit from the rule, a little humility is called for, I think, and, the the players at the table should be given due respect.
Bottom line: Yes. The GM is allowed to use the "Declare a Story Detail" rule in the middle of the scene, and the only reason to do so is for the same reasons any player would: story detail declarations "work to your character's advantage" by definition (FC p. 13).
And this is what they indeed do spend the Fate point for.