The number of steps you take in a mile depends on how long your stride is and how big your feet are. There is no exact number of steps that are in a mile or even in any certain distance because everyone's running form and style and length of stride is different. However, we can give you a few facts (below) that will help you estimate how many strides you take every mile, and what the average amount for runners typically is.
First of all, there are 5,280 feet in one mile. So let's think about this for a minute. We can say, on average, a person's foot or shoe length might be around one foot long (give or take a few inches). So planting your foot on the ground takes up a foot and then when you put down your other foot during your running stride, you typically do not place it right immediately and directly in front of your other foot. Usually there are several inches (if not feet) between your two feet during your running stride. And folks who have larger feet and longer legs and a greater stride length will have even more space between their feet. But for the average individual, there is about a foot to a foot and a half between your feet. So you have to consider the length of your foot (one foot long) plus the space between your feet (about a foot and a half) which totals about two to two and a half feet in your stride length.
If that is about the length of your own stride when you run, then you can estimate that you it will take you about 2,000 steps to run a mile. However, some people have much more space than just a foot in between their feet when the stride. It might be worth your while to measure the distance between your feet during your running stride, and that will give you a more accurate idea of how many strides you will take when you run a mile. But if you are an avid runner already, it might be worth looking to get your watch fixed, or even spending the money to buy a new one. While it is probably not ideal (it is never really ideal timing to have to make a big purchase like that!), if you are a runner, having a decent running watch is definitely worth the investment.