I'm sorry to hear that your fellow ate a dried out Chlorox wipe with bleach.
Chlorox wipes have detergents as cleaning agents and a small amount of bleach. All detergents and bleach are gastrointestinal irritants and can lead to drooling, vomiting and diarrhea with a loss of appetite. But the amount of detergent and bleach on one wipe is very, very little and since it was dried out the hope is that most of the bleach and detergents have evaporated. If he seems to feel fine then likely irritation from those ingredients won't be an issue for him.
Of course the other concern is that the wipe may wad up in his gastrointestinal tract and cause a blockage. If he chewed it up first then this is much less likely to be an issue.
How long has it been since he ingested the wipe?
It may pass but if he just ate it, within the past 2-3 hours, it's worth trying to get him to vomit it back up.
You can give 1 tablespoon of 3% hydrogen peroxide per 10 pounds of body weight mixed with bread chunks or ice cream (no more than 3 tablespoons even if he weighs more than 35 pounds).Then take him outside and run him around to get the peroxide bubbling so he vomits.
If no vomiting in 10 minutes repeat the peroxide dose once.
If he vomits and seems to clear his stomach and be done I would check to see if the pieces are present.
Then after he empties his stomach to ease his stomach and gastrointestinal irritation.
To try and settle his stomach you can try either:
1) Pepcid-ac (famotidine) at a dose of one half of a 10mg tablet per 10 to 20 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one quarter of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours.
These will reduce stomach acid and should help settle his stomach.
You can feed a homemade diet of 1/3 boiled hamburger or white, skinless chicken, all fats and juices drained and 2/3 white rice. Add 1 tablespoon of canned pumpkin, (not pie filling, just pumpkin), to each meal for fiber to help push any remaining pieces of the wipe through. Feed small meals several times a day.
Once the wipe passes and he's feeling better then start mixing in his regular food, adding a little more regular and less bland at each meal. It should take a week to get him converted back to regular food.
If he is vomiting, even after the acid reducing medication is given, refuses to eat or has explosive diarrhea it is best he see a veterinarian for an examination, injectable medications and oral medication like sucralfate to coat his gastrointestinal system or possible surgery if they think he has a blockage.
But if she shows no symptoms there is no need for you to do anything.
If it has been several hours since he ingested the wipe it is too late to induce vomiting. At that point I would feed the bland diet with added fiber to help push the wipe through and watch him for signs of a blockage. (vomiting, refusal to eat, tense, painful abdomen).
Let me know if you have any questions.