Powderfinger:
First of all, good to hear you are over your illness. Good health is the ultimate wealth!!!
Anyone will suffer tremendous atrophy from being inactive for a year. Sounds like you have taken to the bike well and like it.
The problem is that riding at 14-15 mph on the flats won't make you strong. It could even make your legs weaker possibly.
Due to your illness, you have an overall weakened state of muscular strength from head to toe. Cycling is good for your heart and lungs, but riding as you are currently won't help to regain your strength very much if at all.
Climbing stairs is pitting your bodyweight against the attraction of gravity. Cycling on the flats at 14 mph does not. That is why you found it so difficult to climb stairs. You really need a general strength training program to get you back to normal levels of strength. This means lifting weights.
For the time pressed, or those not really into lifting, just do a few sets of bench presses, pullups (not chinups - palm away), and military presses for the upper body and leg presses for the lower body. This would take about 30 minutes.
The pullups will be the hardest to do. Do them. Most people can't do more than one or two, especially elite riders with undeveloped upper bodies. Boxers and kickboxers can easily do 20-30 or more. Start with 3x1 reps to start. It's much better to do one pullup with full bodyweight, than do a lesser weight on a lat pulldown machine. I can't say enough good things about pull-ups. They are very hard to do at first, but the results are well worth it. When you get better at them with time, widen your grip as much as possible, up to at least twice the width of your shoulders. You might have to buy bigger shirts, because your back and shoulders will get wider!!!
Good health for the average person means decent cardiovascular fitness combined with decent muscular strength. If you are not into the "gym experience" you can get stronger by doing bodyweight exercises and climbing stairs or climbing hills on your bike.
Do some bodyweight exercises like pushups, sit-ups/crunches, and pullups for upper-body strength. City parks and high schools probably have a pull-up bar you can use. Check out this book: "Navy SEAL Exercises," ISBN 0-9654093-0-92495, a good book for starting out with bodyweight exercises. Climb stairs or climb hills on the bike for lower-body strength.
Since your legs are really weak right now, it might be safer to stick with the stairs until you feel a little more solid on your feet. If your legs quit on a climb, you might fall off your bike and get hurt.
Recovering from long-term illness requires a rehabilitation program to return the body to pre-illness strength and fitness. The body thrives on hard work, getting strong and fit. Conversely, it falls apart when not used. Your quality of life will improve dramatically if you rehab your muscles from head to toe. Additionally, strength training will improve your posture and appearance as well. Workout for strength and ride for cardiovascular health. You can jog, but after a mile or two you will be reminded of how good it feels to ride a bike. Don't stop riding!!!
Good luck!!!