It's definitely possible to live with a cat even if you have cat allergies. There are several things you can do to reduce, and remove the allergens that cause a reaction.
Step one is always fresh air, ventilation, and filtration. Open windows when you can to get fresh air flow moving through the house, bring in new air and take away the old with dander and other allergens. If you can't have the windows open all the time, like if it rains or gets cold outside, you can use an air filter like Oldcat suggested (Technically you can use one all the time I guess, but come on, fresh air!).
Here is an example air filter. Though I have no idea how good this one is, it has a 4 1/2 star rating.
You can always give the cat its own space so that your wife has places to be safe from any reactions. The bedroom is a perfect place to make into a safe room. You should be able to spend time with your cat whenever you want, but in the case of an allergic reaction, your wife should have a place to get away.
Brush the cat often to get rid of shed hair and dander. Also, brush it outside if you can, or in a well ventilated room. You're brushing the allergens off of the cat and into the air, which is something you definitely don't want.
Furniture and upholstery are allergen traps. You can either clean them often, or you can put allergen protectors on them. My wife is allergic to dust mites, so we have a mattress cover on our bed. I will warn you they make crinkling sounds when you move on them.
They make anti-allergen sprays that are supposed to remove even pet allergens, though I can't say the effectiveness of those, it might be something to look into.
Watch your vacuum. Old vacuums with bags are especially notorious for kicking back some dirt and dust. You can get allergen trapping bags, but newer, bagless vacuums are much better.
Keep the litter box clean. Not only is cat urine an allergen, but cat litter is extremely dusty. You might want to try different kinds of cat litter to see if that helps. Yesterday's News is made of old newspaper, and Feline Pine is made of compressed sawdust.
Keep the cat clean. While cats will wash themselves, they will not get rid of all their dander on their own. Also, cat saliva is an allergen too (go figure). I personally don't suggest daily baths; especially for kittens. But a bath once a week or so would help. Daily wipe downs with a cloth or kitten wipes are an alternative to baths, and can be less stressful on both of you.
- Make sure you use soap/shampoo that is safe for your kitten. I use Dawn dish soap, which is what is used to clean animals from oil spills. You might want to look into something that won't dry out the cat's skin if you plan on giving baths more frequently than once a week. Consult a vet.
- Don't get water into your cat's ears when bathing, as water doesn't drain from their ears like it does in humans, so it will get infected, even cause permanent damage.
Finally, I would suggest you get treatment -- Even if it's just taking antihistamines to reduce the allergic reaction. There's also the option of getting an allergy test to tell what specifically you're allergic to. That can help you target the allergens more effectively.
You will want to speak to a doctor about immunotherapy for the details but I can tell you this much: It will not cure allergies. What it does, is it gets your body used to the allergens. Desensitizing your body to the reaction and lessening the immune system's response. Your wife still might have allergic reactions after the treatments are completed, but the point is that they're so minuscule she won't hardly notice.
First thing to do is understand what an allergy is. At the basic level, an allergy is the body reacting negatively to an otherwise harmless substance. What immunotherapy aims to do, is expose the immune system to the allergen in minute amounts at first, then slowly increase the dosage until you no longer feel the effects.
The process does take time, so don't expect it to be done anytime soon. The treatment goes something like this: You will go in once a week for a mall dose at first, then once every other week, then once a month, then once every few months, then once a year, then once every couple years. The dosage increasing by a fraction each time. Because there are so many treatments, it can be fairly costly, but I cannot give an estimate with any confidence. I will say immunotherapy has been shown to be an effective method of managing allergies. If you and, more importantly, your wife are patient enough, it will help her manage the allergic reactions.
See more at: The American Academy of Allergy Asthma and Immunology