Well, in that case, I assume that they've worked out a "low energy consumption" flight plan...
Assuming that's true (which makes sense) it would also rather suggest that they would have very little fuel left after they returned to the planet. Otherwise Spock would surely have increased the lead time the ship had on the asteroid. Which he doesn't:
SPOCK: ... that asteroid will be four hours behind us all the way.
That doesn't even address the problem of where they're going to fix the warp engines.
SCOTT: The only thing that'll fix these poor darlings is the nearest repair base.
How far away
is the nearest repair base?
Well... pretend "Star Trek is real."
The Enterprise arrives at the planet. Kirk is rescued. The asteroid is deflected. The Enterprise enters orbit, and makes a call to the nearest starbase.
That starbase has at least one "tender" ship on call at all times. The tender takes roughly a week to prepare a payload of the required hardware to repair the Enterprise, then sets sail. Let's assume that it takes the better part of a month for the tender to arrive.
During this period, the Enterprise continues to study the local system, the transplanted "indigenous" population, and of course the "Preserver" technology on the planet. A month is barely enough time to scratch the surface on this sort of an activity, but the Enterprise is outfitted to do a reasonable job on this task.
(Presumably, Marlena McGiver's replacement would be doing something at this point... there's a reason that the ship has a historian, after all!)
The tender arrives and the Enterprise sees her major systems repaired. Once the repairs are done, the tender returns to base. During this time, a dedicated survey vessel may have arrived, but in any case, it's most likely that the Enterprise would stay in orbit until a survey vessel was on location, then would receive orders to set sail for their next mission destination.
That's how I envision things.
What is a "tender?" Well, it's a ship dedicated to repairing other ships. It would be something somewhat akin to the F.J." transport tug" in some ways... it would carry a massive cargo bay capable of carrying major ship components which might require replacement. It might even be able to carry entire nacelles, though I doubt you'd see that happen very often, as that sort of thing really requires a drydock as far as I'm concerned. It WOULD have the ability to fully support the serviced vessel... ie, the vessel being serviced could shut down any or all of its major systems entirely, and could "slave" to the tender (anything from power to life support to command and control systems). It would carry a large contingent of "work-bee" type craft, and many, many "construction spacesuits" as well. At least half... probably more... of its complement would be repair crews.
ANYWAY... that's what I envision happening. But since that would be dull to watch, that happens between episodes.
