I'm not sure, but I'm fairly certain it's longer then a day.
A week minimum, maybe 2.
At least, that's how long I've noticed between the special reward going live and the notification in the launcher telling you it's your last chance to get the special.
I think you're right about the 2 weeks. That's been the average they've given in the past, so I'm expecting this one will be available for about 2 weeks after the episode airs as well.
I'm kind of looking forward to the "re-run" series dstahl's posted about, mainly because two of my friends stopped playing STO around the time of the first two series, and now they've come back and will be able to claim their Breen officer and Devidian cane.
(Plus, I might actually level up my fresh-out-of-the-tutorial Ferengi, and have him run through them as well, heh.)