Disclaimer: I am not a doctor, nor a dietician. I am the type of person who finds it easy to gain weight or lose weight if I put my mind to it. Interestingly, I have helped friends lose weight by doing what I do. Y(W)MMV.

Depending on the size of your wife, I think that 1200 calories per day is too low. In fact, when I have crunched the numbers with nutritional software (http://nut.sourceforge.net/ absolutely worst UI in the history of the universe, but otherwise excellent), it is really, really difficult to balance nutrition on less than about 1600 calories a day. Potentially, you could do it with supplements, but eating a realistic diet that you can maintain is very important for improving your lifestyle (in my opinion).

My advice is to up the calorie intake in order to improve nutrition and correspondingly up exercise levels. You say she has several hours of exercise per week. I assume that means that she is exercising a few times a week. It would be much better to exercise 6 days a week (taking one day off for rest).

Intensity of exercise is important. If the intensity is too high, you will run out of sugar. If your goal is long term energy expenditure, then lower intensity for longer times is much, much better. Unfortunately, I don't have time to find you some links, but do a google search for graphs of percentage of sugar vs fat against exercise time.

It depends on the person (especially how fit you are already), but essentially the body starts out burning sugar and slowly starts using fat in preference. Generally speaking it takes about 30 minutes before you are burning 50:50. So if you do 15 minutes of intense exercise - to the point where you can't continue any more - you will have depleted your sugar resources. This impairs your body's ability to power itself for the rest of the day. Much better is to lower the intensity so that you can do 30-60 minutes of activity. This will spare the sugar in your body, while burning calories (from fat). Also, fat has nearly a 4:1 energy density advantage over sugar so you should find that the exercise is more enjoyable and your performance is much higher. In fact, a good 20 minute warmup to get the blood flowing and the muscles working, followed by a good workout will do wonders for you.

The level of intensity is quite important (and is something that I personally have difficulty with because I always train too hard). For the past 10 years or so, I've been doing cycling. At first I was riding hard for about an hour every day, to the point where I collapse at the end of the ride. My wife wanted to ride with me, so 3 or 4 days a week, I had to go at her pace. We still did an hour a day, but at a really slow pace. After a month or two, I could not believe the difference in my performance. It just sky rocketted. There are many scholarly papers on the topic which you can search for, but the old advice of doing base miles seems to work extremely well (at least for me). The idea is to maintain a level of exertion where you are slightly out of breath (have to breathe through your mouth), but that you can have a conversation easily. As far as I can tell, it literally doesn't matter what exercise you do, as long as you keep to that rule.

Obviously, 6 hours of exercise a week is a lot and it is hard to organise your life around it. However, 6 days a week is much, much easier than 3 or 4 days a week because it forces you to not be flexible with it. You must plan for it and make sacrifices. In my experience (you might be different), trying to add exercise (or anything else) to your life while avoiding impact to your life just means that you will be inconsistent. Take the hit up front and make it a priority. After that things just get easier. The other way around just stays difficult for ever.

Another important thing is to pick something that you enjoy. I used to run and if I took too much time off, I would always find that the first 2 weeks of getting back to it were hellish. After that, I would fall in love with it again, so sometimes you just need to persevere even if you don't think you are enjoying it. However, with cycling, it doesn't matter how long I've been away, I love it from the first down stroke of the pedal. If you don't enjoy what you are doing, keep searching for something that you do enjoy. It's going to be a big part of your life.

The number of calories you burn depends greatly on the sport you do, but I think a good rule of thumb is that you will be burning about 5-600 calories per hour. My wife just handed me one of those 1/3 sized cans of Pringles chips. There are 61 grams of chips in the can comprising 316 calories. I can easily eat this can of chips without even remembering I did so (wiping out half of my hour of exercise). If you always look at labels and think 100 calories is 10 minutes of exercise, it hits home. In fact, go to a Seven Eleven and look at the labels on things. Can you even buy a snack that has less than 100 calories?

As for food composition, I'm afraid to say that I think people are over-optimising on the wrong thing. Whether or not low carb is effective, I don't know, but none of the people who have followed my advice has ever failed to lose whatever weight they wanted (see disclaimer below!). If you must obsess with what you are eating, I would err on the side of reducing fat. Again, the calorie density thing means that if I'm eating lower fat, I get to eat more volume. For exercise, I firmly believe in sparing sugar, and I can't imagine any low carb diet would perform well for endurance sports.

One last piece of advice: don't try to lose weight too fast. Absolute max in my opinion is about 1 kg per week (about 2 pounds). Honestly, I would dial that back to 500 grams per week because it just seems healthier from my experience. If you do this for only 1 year, you will lose between 25-50 kg. That is a crazy amount of weight (and you will already have a lot of loose skin that won't rebound in that amount of time!). If you think, "I don't want to wait a year", then my experience is that you will fail anyway. It's not about the goal, it's about the journey. Without experiencing that journey, the goal will disappear. Find a way to enjoy it -- it's the only way.

One more disclaimer: if you need to lose more than 20-50kg, don't follow my advice. Seek a professional and get qualified advice. Random people on the internet are not qualified to help you (myself included).