A stationary bike, like any exercise, can help you to sleep better. The physical activity it provides stimulates and stresses you (in a good way), and your body will compensate for this by sleeping better. The best exercise for sleeping well is one that provides you with a good workout without overdoing it.
Exercise that does not challenge you enough will not help you sleep much better (although anything is better than nothing). And exercise that is too challenging on your body can be dangerous to your health and may cause you to sleep poorly.
A stationary bike can be appropriate for anyone with any degree of physical fitness. However, people who don't have much history of exercising and are not in good physical condition often find that a stationary bike suits them. That's because they can do it in the privacy of their own home where they will feel more comfortable. Also, it's fairly hard to overdo it on most stationary bikes, and they are much safer than regular bikes.
A Bored Mind Does Not Sleep Well
I've noticed that stationary bikes are bought with the best of intentions, but they often end up sooner or later in the garage, never to be seen or used again.
The main reason for this is that stationary bikes can be, by themselves, boring to use. Unlike a real bike, a stationary bike lacks the sensations of speed and changing scenery that help keep things interesting for your mind. This is a problem because, just like your body, your mind should be challenged and stressed (in a good way) throughout at least part of the day. When it is challenged, it will recover from this by inducing better sleep at night.
Challenge Your Mind as You Pedal
To ensure that a stationary bike stimulates both your body and mind, you don't want to just pedal away in some room with your mind idle and bored. Instead, watch an intelligent show or movie on TV while you cycle, something that makes you think and watch actively instead of passively. In other words, stay away from virtually all sitcoms(!). Another option, of course, is to read something. Again, the best choice is something that your mind
A stationary bicycle can be a good workout for your body. And the time you spend on it can also be exercise for your mind. By challenging your body and mind throughout the day, not just one or the other, but both, you greatly improve your chances of getting a good night's sleep.
And you'll also improve the chances of the stationary bike staying out of the garage.