Oooh, Saratoga. This is my area of expertise.
Righto, in order:
1. You want her taking that torpedo hit. Any potential bonus at Coral Sea (doubtful, IMO, looking at deployments) is balanced out by the fact she got a badly needed refit due to the damage. If she doesn't take that hit, she's not getting her full refit until one of two things happens. The first Essex-class come online, or she takes another sort of hit...that may not be as survivable.
2. It is possible to expedite her repairs or working up, though, this leads a bit into Point 3. It's not as useful as one could think.
3. Sara was already at Midway. Spiritually, at least, through her planes and pilots. Her most veteran aircrews had already been pulled and stuffed on Yorktown. This was both to refit Yorktown's air wing, after the losses at Coral Sea, and because Nimitz figured 'better to have one carrier with a full wing, than hope another one works up in time'. That's a horrid oversimplification, mind, but it gets the point across. Yorktown had a motely mix of her own, survivors from Lex, and Sara's best. Which leads into...
4. She has relatively few planes to provide. Sara had, give or take, about 40 planes with pilots ready to go. Most of those were Dauntless, which are certainly helpful, though only if they can find the Japanese carriers. If Sara is with Yorktown and launches a coordinated strike, this is a decent possibility. If she is arriving after Yorkie, then she's going to risk having Hornet-syndrome* and then it's just having more planes that don't do anything. Furthermore, only about...a dozen or so of her planes (I believe it was fourteen, to be exact) are Wildcats. If those aren't sent to escort the bombers, it's not a huge increase in CAP. A noticeable one, to be fair, but not a Fighter Carrier. Which is where...
5. The vast majority of the planes she has are in crates. In the best case, you'd need a day or so to get everything put together- Sara actually has the facilities aboard to disassemble and reassemble planes, no need to be in dock -and can have another sixty (give or take, again) planes aboard. I say aboard because she can't actually operate all of those, not reasonably. Fly them off to another carrier, sure, but not from her own deck in a reasonable manner. This presuming she has enough pilots for all of them.
(The loadout that CalBear mentions is after she arrived at Pearl and was explicitly loaded up with replacements for the other carriers. I'm looking at what she left California with, instead. The most likely scenario, in the event she finishes refit early, is to have a mix of the two. What she carried to Midway IRL was not a proper combat load, so much as replacements. What she carried from California is similar, though if she gets there just barely in time to uncrate and put together her planes, she has to use what she has)
6. In the absolute best case scenario, you've got another 20-25 fighters and about 50-60 assorted bombers to throw at the problem, IMO. Assuming she operates with Yorktown, her refit will provide more AA fire. Furthermore, the strikes on Yorktown by Hiryuu used relatively few planes. Because there were hardly any left for the Japanese to use. If those planes are split, then it's entirely possible neither carrier is hit. Between Sara's refit and the extra planes. I-168 is not going to get lucky here. She got hilariously lucky IRL in finding Yorktown being salvaged and sneaking close enough to fire. That's not liable to happen with two carrier's worth of escorts and two (presumably) functional carriers.
(Also, there's the chance the strike never happens anyway, if Sara's planes don't all target one carrier and hit Hiryuu. Assuming that E and Yorktown hit the same targets as IRL)
In a less favorable scenario, you're still talking about another carrier worth of planes. Unless they screw off and miss the Japanese entirely, that's another full strike wave. Lots of potential there. If she is not with Yorktown, then one of them may still be hit without supporting each other. Sara can take the hits a bit better, but it's hard to say at that point, exactly what happens.
In the worst case scenario, the Japanese get lucky and the Americans don't. Say that you don't have Arashi lead the way to the Japanese carriers. Say that the Japanese find the Americans first. At that point, having one more carrier won't do much good. It's a bit more likely that at least one carrier survives, though, compared to if the Japanese pulled that off against the three Yorktowns instead.
I could probably go into more detail, but it's late and I'm tired from work :V
*Hornet-syndrome is referring to how Hornet's CAG lead the vast majority of her strike force in the opposite direction of the Japanese, and stubbornly refused to turn around. This is why most of her planes missed the battle entirely.