That's a lovely job.
I made a box-jointed octagon early last year - it was hard!
First off, is it piano hinge you've cut to length? If so, do you have offcuts to experiment with? I assume you want to fix the existing hinge rather than fit new one(s) as that would, er, mess up the finish.
I'd cut a couple of new hinges from whatever is left over, assuming you can match hinge and screw hole spacing.
I'd test fit to two long bits of scrap softwood, so you can see if you;re making an improvement. Test for wiggle/play, mark the amount on the scrap stock and un-mount the hinge.
Then, I'd use a smooth-jawed Mole wrench REALLY CAREFULLY to see if you can pinch up the hinge a bit to take the slack out of it, one bit at a time, starting from the ends and working inwards.
They're usually made from brass-plated steel. The better ones are milled brass (as fitted to real pianos), but I doubt you can obtain that any more. You should be able to gently squeeze where the sheet has been rolled over to make the pivot. it might be worth putting aluminium tape on the Mole jaws - ought to allow you to still get good grip but not mark the work. Mole grips aren't parallel jaws, and close to an acute angle, which is what you want. Don't overdo it and prevent the hinge from moving entirely!
If it works, you have a hinge to replace the one fitted. If not you've not made the box worse.
E.
PS: I was wondering about lubrication - it might be worth warming the replacement hinge and running some molten candle wax over it, so that it penetrates. Wipe/scrape off any excess, and what's left in the hinge should help prevent rust and keep it slippery in a reasonably clean way (shouldn't pick up dirt, nor mark anything it comes into contact with). Might not work though.