Shiny.
You don't actually need much here. Let's say that Kaylee's thruster is an outer one. You don't even need to eject it, just have it rendered inoperable. This means that firing all other thrusters would imbalance the ship and throw it off course. So it's not just the impact of losing one - you actually lose at least two of the propulsion units. Let's say this cuts the ship's effective top acceleration by around 20%.
Now, in order for them to move close to the asteroids (in order to mine them) let's say they had to cut speed and move off course slightly. These aren't major adjustments, but they'll be significant, because losing 20% of the ship's max acceleration after that means they can't hit their next molecular cloud target directly. Instead they can just skim it, and that will cut the amount they can harvest next time by over 50%. And this will cause the delay because they're forced to change their route past that point, since they won't be able to make their next target after that. They have to pick a new cloud and kind of work their way around.
As for the accident itself, it depends on your exact propulsion methods. I would have the asteroid destroy a key, irreplaceable component. For example, if she's running a fusion drive, maybe the impact destroys a perfectly formed containment unit, something that can only be built in a specialized factory. It's well beyond the generation ship's means to replace or rebuild, and can't be jury rigged.
As to why they decided to send the ship out with components it can't replace - maybe they just didn't have a choice. Or possibly the ship did have some spares, but they were used on previous fusion units that failed, or maybe it turns out the spares were built by a low bidder and have hidden faults.