Our cat is 20, but he would have died at least 5 times if it weren't for a trick we learned for helping cats survive one of the most common killers: bouts with severe diarrhea and/or vomiting that are so bad a vet will often recommend euthanasia.
The trick:
First of all, the best solution is prevention. A cat's own hair is frequently the cause of (or major contributor to) the above-mentioned illness. If you shave your cat a couple of times during the summer when they're shedding, there's a good chance you'll prevent the illness.
The trick is to get a syringe from a vet or pharmacy (or online) and remove the needle and use it to squirt little bits of room-temperature unflavored Pedialyte (found in infant section at store) into your cat's mouth. Start with 1/4 of a syringe every half hour. If they can keep that down, try 1/2 syringe every hour instead to keep from interrupting their much-needed sleep too frequently. If they can keep that down, try a full syringe every hour. If they can keep that down, try two full syringes every two hours.
Your goal is to find out how much Pedalyte can be given at once without triggering vomiting, by starting out small and going larger (not the other way around).
If you get to a point where you're managing to give quite a bit of Pedalyte in a single session without triggering vomiting (eg, three syringes, once every three hours), but you trigger diarrhea instead, you'll need to reduce the overall amount of Pedialyte you're giving, or keep the amount the same but spread it over more sessions (ie, increase the session frequency but reduce the amount given per session).
Keep in mind that this is a 24-hour process, so you'll either need two people, or you'll need to wake set an alarm to wake up for every session.
If you manage to get things going well (eg, 72 syringes in a 24-hour period without vomiting or diarrhea) you can try replacing some of of Pedialyte (eg 14/ syringe per session) with kitten milk sold in the pet section at store. If you use powdered kitten milk (which is usually the better choice), hydrate it with bottled water, not tap water.
If this goes well for 12 hours, replace more of the Pedialyte with kitten milk.
Ideally, you want to reach a point where you're giving two syringes of kitten milk for every one syringe of Pedialyte, and, if that goes well for 12 hours, replace the Pedialyte with bottled water.
If things ever start going bad again, replace some of the kitten milk with water. If things start going really bad, start back at zero with small doses of Pedialyte again, and try working your way back up.
Every cat is different and it's up to you to find the right balance for your cat. Keep in mind that some cats will start vomiting if you put too much time between sessions (eg, 4 hours) because their own stomach acid starts to irritate their stomach.
If things go well for three days, you can try feeding a couple spoon fulls of plain ground or blended chicken breast to see what happens.
To make it it, you simply boil the chicken breast for about an hour (use just enough water to cover the chicken), let it cool then grind the chunks of chicken in the grinder, then pour the ground chicken back into the water you used for boiling and stir it together.
If you don't have a grinder, you'll either need a really powerful blender or you'll need to chop the chicken into little bits before blending. Use the water from the pot for your blending.
You can try chicken baby food if you want but it's far more expensive and usually has additional ingredients you don't want.
Typical time for cat to be out of danger is between 4 days and two weeks.
If it takes two weeks, your cat will have lost most muscle and will look pretty scrawny, but blended/ground chicken will put the muscle back on pretty fast. Once they're eating well, I usually add a spoonful of canned cat food to each serving of blended/ground chicken. Chicken breast alone doesn't have all the nutrients they need.
Notes::
The syringe size I'm referring to is about 12ml. If you can't get your hands on a syringe for some reason, check the infants section at the store for a dropper of some kind.
When you squirt fluid into kitty's mouth, do it sideways, in from the side of their mouth and down onto their tongue otherwise you risk them breathing it in which would be really bad. Only squirt little bits at a time.