I think Buh nailed it. It's what I was trying to explain, but as usual, better.
It
lasts
the entire duration of the attack - in fact the entire combat round, if you want to get technical, but if it's used she gets stressed and it can't get used again until that stress is cleared. As Buhalin said - if you could do some cross faction craziness where you had her with Captain Yorr (as an off the wall example), where he could take that stress token FOR her, she could in that example use her ability on both attacks of the cluster missile.
She adds the die. Takes a stress token. Yorr intercepts and takes the stress token for her. She now has no stress and is free to use her ability a second time.
I think a relevant part of the FAQ is on page 6, regarding cluster missiles triggering Gunner or Luke.
"Each attack granted by Cluster Missiles is distinct, so either attack can trigger Gunner/Luke Skywalker."
Also important to note that if you fire CM, miss, decide to trigger gunner - you are not allowed to resolve the second cluster missile attack. It is overwritten by the "You may make no further attacks this round" text on Gunner/Luke.
I think that's a pretty important precedent for how Cluster Missiles would interact with Jan, as well.
Which I now see Ken just very clearly explained RIGHT above this. I have literally no idea how I missed that post, so all due props to Ken for
that answer first. I wish I could claim he snuck it in while I was typing this, but he posted that an hour ago.