I have added a kitten to a much older, set-in-her-ways cat, but not with also 2 dogs. My youngest DS begged for a dog, but with our busy lifestyle, we knew we didn't have the time. After several dead goldfish, we grudgingly agreed to a kitten--another female kitten to go with our older female kitty. Turned out the shelter was wrong about the gender of the kitten my DS picked out, and it turned out to be a male! oops!  DS was already attached, so we brought the kitten home. My one-pet-household for her entire life older cat was NOT happy at first. She hissed at the kitten and made sounds I'd never heard before--I'd describe between a "meow" and a "growl." lol. My older kitty would not eat for a few days and would just sit with her ears pinned back. I finally bought some nice canned food (her favorite) and loved on her and she came around to eating again. And after a while, she finally got used to the kitten, who's now almost 3 years old! He gives her grief sometimes (wanting to play, which she clearly does not!) but for the most part, they get along ok. I come home sometimes in the middle of the day and find they're both sleeping on the same bed, so I think they're good company for one another. And maybe your existing older cat would find another cat a good ally against the dogs, which currently outnumber your kitty. I'm sure it's workable. And it sounds like if you don't adopt the kitten, you're going to feel awful, so maybe you should just go for it!

I have done it. I had an older cat and two dogs - both high prey. One of them "treed" a couple of half grown kittens and DH brought them inside. I knew they belonged to a neighbor who lets her barn cats breed indiscriminately, so I decided to just keep them. (My whole life cats have found there way to me - I have had as many as five and have never gone to a shelter - they just find me.)

The older cat was not happy with the newcomers - but she eventually adjusted. I started out keeping them separate, the kittens in a room and her having the run of the house. I then reversed the situation. She put them in their places pretty quickly, and they learned to leave her alone.

My older dog knows that you do not chase the cats in the house -  even if they run, so she wasn't a problem. The younger dog would chase if they ran. I had several "come to Jesus" discussions with her and she learned not to chase, even if they ran. I tried not to give her the opportunity - for several weeks the kittens and the dog were never left alone. I kind of went overboard in the beginning, she got so she wouldn't have anything to do with them for fear of getting in trouble.

In your case I'd start by introducing the new cat to the dogs while she is in a carrier. And keep the introductions to the dogs separate from the introductions to the cat. You don't want them to excite each other.

Gary kitty was around 15 when we brought home a puppy. She hated him for a week, ignored him for another, but by the third week she was sleeping next to him in his bed. She was 17 when we introduced puppy #2 but by then she really didn't care. She lived to be 18 and hated most people and other living things. But I'd bust her in that puppy pile every day until her last. ❤

3 dogs, 2 cats.

Great advice above.  My mom taught me to always feed the cats first so they are ‘higher up’ in the pet hierarchy.  I have never had a problem introducing cats into a household.  Just take it easy getting everyone to settle in and let the dogs know they are ‘under’ the cats.

So not with cats, we may introduce one in the future (once we're in a bigger house, we have two turtles and three dogs and they're inside babies... we go to the dog park quite a bit, but we def need more space for any more furbabies!).

But the puppy that DH brought home due to his coworker "just going to be rid of her since the shelter is full."  She's been introduced swimmingly. At first they definitely weren't sure. My dog was beyond intrigued, kept by her side, had a VERy protective sense, but was extremely annoyed if he got touched by her. Dh's dog was growling at her and showed some teeth, I thought for the first two days I was going to have to find her a new home because i didn't want him to snap at her. 

BUT, I made sure they weren't left alone for a few days, started with short spurts of home alone and after just a couple of days they get along GREAT. All of them are playing together, they're gentle with her. Honestly they're being super great big brothers. I walked in the other day to her laying on DH's dog and they both looked content as could be.  She follows them around everywhere and tries to copy them (which is great, has made potty training a breeze and she can already sit and lay down!)

I think new animals can be introduced into the home Just means some diligence on your part during the adjustment period.

Don't give up on this kitty. If you know in your heart you can give her a good home then just do it and take her in. 

There is some really good advice here on introducing them with the dogs.

I've taken in homeless pets and feral cats all my life, including feral cats that people said would never be tame. I can personally tell you, the feral cats all became the best, loving, tame pets I've ever had, and they were indoor-only during their lives with me and DH. They all got along, but it took time and patience.

If any of this helps you out:

Cats feel safe up higher than ground level. We have a multi-tier cat stand. The tallest curved cat seat on it is about  6 ft. up. We have only one cat now, and he hangs out on the top tier a lot, especially when he sees foxes, coyotes and dogs outside. So, if you have a place that the new kitty can access easily up high enough so the dogs cannot get to her, she will most likely feel safe there. 

When we introduced our current feral cat, now a big love bug, to our other two cats, (now deceased), we took a large piece of plexiglass and made it into a temporary door by duct taping it on one side-like where the hinges would go and keeping new kitty in a room on one side of and the resident kitties on the other side. That way they could all see each other and touch their noses to the plexiglass without any bloodshed. This also gave resident kitties run of the house while they got use to new kitty. It took a few weeks. At first there was hissing and growling through the plexiglass, but no one got hurt. After a week, the hissing decreased and they became more curious of who this new cat is. We then removed the plexiglass and let them be together while monitoring every second at first. This was a tip from a veterinarian and it worked so well.

also, many years ago, I did the same thing with a screen door on a room with two new cats I took in. That worked well also.

I also find if they all have their own food and water bowls and pans, there is no need to fight over food or water. 

And..toys, if they all have enough toys and scratchy posts to use, there is no fighting for some. Also, neutered and spayed cats and dogs are much, much less prone to fighting than unneutered.

And..the best thing that came out of all this. Just seeing the bond some of them formed with each other while becoming best friends and knowing they were all loved, safe and happy was well worth the time, stress and effort in gradually introducing them to one another.

At first, I almost gave two of the ferals I brought in to other homes, as I wasn't patient enough to allow them the time they needed to get along with the spoiled, resident cat. But, after giving them a few months, they all bonded and resident cat and one of the ferals became inseparable. 

I know many people who have dogs and cats in one home and growing up I always had many pets. It can be done and takes a little time, as you know, but knowing the lives you save make it so worth it.

Best of luck to you with the kitty.