Hi Monica! Some great questions and you're right to be concerned. I actually suffer from some serious bronchitis and asthma so I know exactly what it feels like to try to work out and getting out of breath. I can tell you that there are several people I work with across many disciplines who have chosen to start martial arts to help them with their breathing.
Aikido and Jiujitsu are pretty similar so either one you start with won't be a bad option. I honestly believe that for a beginner, the key is more who the teacher is than the style itself. Do you have either styles taught near you?
I don't think there's anything that you can go "wrong" with. You sound like you've recovered quite well from your bronchitis if you're able to get on the bike and other activities. I would really take some time to visit some schools to find out.
I had promised to do some articles this month about training so I might as well go ahead and start with the first one:
Questions to ask when you prepare to visit a school. Might help you out. Please let me know if you find this helpful.