I'm an asthmatic, so I have similar issues and it was much much worse when I was really out of shape. For me the best answer was to start off slow, do what you can today, don't worry if it doesn't seem like that much and then tomorrow or the day after increase by a tiny bit. So if you can tolerate the treadmill for 6 minutes today, try 6 1/2 or 7 the next day, so forth and so on, at the end of a month you'll be amazed at how much longer you can tolerate to workout. Also I highly recommend doing calisthenics, building muscle helps your body become more efficient at burning calories and fat. You really don't have to run a marathon to put in a really good workout, you also don't need to be at it for hours to get a good amount of benefit. I used to take a "Toning" class at the fitness center, it only burned about 1/2 the calories that the aerobics class did, but I always felt it the next day. Not so for the aerobics, after the first couple of weeks my body got used to that and although I'm sure it burned calories I don't think it built much in the way of muscles. Another thing, don't knock walking for exercise, or at least don't knock it until you've tried it. In my life I've known some people in really fabulous shape, and every last one of them walked on a regular basis, and most of them did little else in terms of exercise.
For workouts that don't get you gasping you might try yoga, walking (there are some decent videos by Leslie Sansone for bad weather days), core strengthening and weightlifting. For Christmas my husband bought me "EA Active 2" for the Wii and it's pretty decent, I don't gasp through it, but I do break a sweat and get muscle soreness from it. And I can go full out on the elliptical for an hour and never get so much as a leg cramp from it.
Starting weight: 244.6 lbs. 10/01/09
Pounds lost: 80.6 lbs
Current weight: 164 lbs
Goal weight: 120 lbs
Weight to go: 44 lbs
Goal Date: 1/1/13