I also have exercise induced asthma. What i have found best for exercise FOR ME is dancing and rock climbing and hiking. I'd been a runner for most of my life, running several miles several times a week, but it was like torture, my lungs would feel as tho they couldnt handle it. I can totally relate to your struggle. As far as I know swimming in chlorinated water is no good for asthma, its one of the things that I attribute the cause of my asthma too.
The hardest part about exercise for me has always been STARTING and keeping a routine. Sounds like endurance may be a problem that you have and the only way to get around that is to start slow and work your way up to a healthy level of endurance that would allow you too get in some cardio without the pain.