Simply put, getting into Big Dig or Breaking The Planet on KDF is impossible. I try to remember to queue myself up to at least Big Dig whenever I'm on my Klingon, but it just never runs anymore, at least while I'm online. I don't really like Breaking The Planet, but have been trying to queue up to it aswell, with no luck. Thus wasting 2/3rds of the fleet action dailies.
To fix it, I suggest lowering the minimum number of players required to start the mission. Players probably don't bother queuing because they know its not going to run anyway. Feedback loop: Players don't queue because they think it won't start, and it won't start because not enough players queue. Not being able to see if there are other players in the queue doesn't help, either.
The queue window says the minimum is 12 players, which is way too high. I think it used to be 8, which is still high. In a private queue, you can start Big Dig alone. I've even completed it alone like that, just to see if it was possible. It takes an impractically long time, but it can be done.
However, starting alone or with just one or two others shouldn't be a problem in a public queue, as more players would be able to join in during the mission. And more players would be likely to queue if there was less waiting involved.
Exactly how low the starting limit needs to be depends a lot on just how empty the queues are,, something us players aren't allowed to know anymore. But 1 to 3 players should be enough, certainly no more than 5.