If you eat healthily and exercise daily, then you should be losing weight. I think you have to figure it out through process of elimination.
Are you really eating healthily? I sometimes kid myself some meals I enjoy are healthy, because they're not out a packet or deep fried, but I know deep down they're not really healthy. It's surprising how much butter on bread or mayonaise on salad can all add up as well. Are your portions the right size? Are you having too much of a particular group eg carbs? Do you snack at all (I'm not an advocate of snacking, but it's not a bad thing if you eat a small amount of the right thing). Fruit contains a lot of sugar, so excessive consumption may lead to weight gain. Are you drinking a lot of fruit juice/smooties/other high in sugar drink/alcohol? I think using something like my fitness pal or similar to log every calorie you consume for a week would be the best place to start. If it comes to more than you were expecting, you may have your answer. If it comes to less...
Then the next thing would be to rule out medical issues, eg thyroid. If you do have something like thyroid issues, you have my sympathies, but don't be defeated, you can still lose weight, it will be difficult, but it can be done.
With the exercise, have you perhaps reached a plateau? Are you really pushing yourself? You need to pick an indicator and then work to targets, continually moving them. For example, when I started working out, I did 15 minutes on the bike. When I felt ready for more, which was approx every 10 days, I increased how long I cycled by 5 minutes. When I got up to 30 mins, I kept the time the same, but every week or two, put the intensity level up. When I reached the highest intensity level, I recorded the distance I cycled in 30 mins then I tried to knock 10 seconds off my time each week.
Doing things like that will help keep you working hard. It's important to only pick one at a time though, eg only pick one out of increase the time cycling, increase the distance, increase the intensity level, try and beat your previous time at a set distance, as it's easier to keep track of your progress and you don't want to set unrealistic targets.
I would keep a brutally honest food diary and push myself while using the bike for a week (I was told you should be "out of breath" but not to the extent you can't speak). Then you're in a better position to evaluate whether you should see a doctor.