In general, if jars are improperly processed or don't seal, you reprocess them exactly the same way you did the first time. This doesn't depend on the original recipe; you just have to do the exact same thing over again. In your case, since it sounds like your original process was hot pack, you would have to open the jars, dump out the salsa and reheat it, resterilize the jars, and reprocess.
But in your case, unfortunately, it's too late. If you didn't process it long enough the first time, you have to assume that it wasn't sterile, and treat it as equivalent to not canning it at all. Yes, probability-wise, your situation is safer than that, but you have to plan for the worst case, because you have no way to tell. Something may have grown in there. Canning just isn't a good place to mess around with safety.
And before someone chimes in and says it, yes, if you're the kind of person who eats things that have been left out way longer than is reliably safe, you could save it. It'll work out some fraction of the time, so there are plenty of people out there who can truthfully say "I've done this and never had a problem." But it could also get you sick, so I can't recommend it.
Yes, you should fully process your salsa, even if you got the "hot jelly" seal just from hot packing.
The reason for this is that part of the processing is to ensure that the entire contents of the jar, all the way through, is at a high enough temperature for a long enough time to be safe as a shelf-stable product. This may or may not already be true from the initial cooking down, but in these matters, you absolutely positively want to be sure.
Best Answer
Per Joe, the answer about jam jars that didn't seal applies to you as well.
Given that the salsa has now sat out all night unrefrigerated, it would not be safe to re-do the canning process. If you had checked them immediately, you could probably have used new lids, cleaned the rims of the jars, and hot-water-bathed them again. But not after 8-12 hours at room temperature. If they have very high acidity (PH less than 4.0) they might be fine, but why take a chance?
Instead, I suggest putting those jars in the fridge and eating them yourself within the next few days, checking them for bubbles that indicate that the salsa is going off.