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"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 p74, 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)

"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)

"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 p74, 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)

So: when does a GM pay fate points for creating story details (if ever)? When do they simply narrate? Certainly there's a difference.

"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 p74, 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)

"So: when does a GM pay fate points for creating story details (if ever)? When do they simply narrate? Certainly there's a difference."

"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 Myguy has Supertool N right here and now! Now to use it to disarm the trap which PC X just set for me." 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.

Bottom line: the GM is allowed to use the "Declare a Story Detail" rule, 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 spend the Fate point for.

"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.)

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.