What kind of exercisesis should I do if I have a broken disc in my lower back and I have asthma so I can't run and im trying to get in shape?

Broken disc? Depends on you actual diagnosis, wether you perhaps have a bulging disc, or a herniated disc, and wether or not the protruding disc material is intruding on any neural structures or your spinal cord. Discuss your MRI results with your dr. Physical therapy may be an option, and if so you physical therapist besides stretching and strengthening exercises would be able to recommend appropriate cardio.
See MD. In general, asthma should not be a limiting factor in an exercise program. The goal of asthma therapy to allow a patient to do whatever they want whenever they want to. If your asthma is stopping you from exercising speak with your md. With regards to your back problems speak with you orthopedists.
Swim, elliptical. Swimming could be a great option for you. Also, elliptical is low-impact and puts the spine in a neutral or slightly-extended position. Light walking in warmer, more humidified air also an option. Of course, you have to monitor your symptoms and individualize your exercise. If anything causes worsening of your low back pain or asthma symptoms, then discontinue it.