It isn't easy, but I think you'll feel that once you're exercising frequently, that it will become much easier to add time to your workouts.
When I first started going to the gym, I thought I would just die after 15 minutes. I remember watching people run for 30+ minutes on the treadmill and wondering how they were alive at the end.
It wasn't too long, though, before I could match their time (maybe not the level of exertion, but that, too, came later).
I recommend that you go in with the goal of using the bike AT A MINIMUM of 3x a week, and if you do more, that's great. Make 3x a week, 15 minutes the absolute minimum that you will do, and I think that you'll find yourself on the bike more than the minimum.
Keep us updated on your progress!