$10 LED flashlights are generally keychain lights that are simply not large enough for general emergency use. Most are not very rugged and few are even moderately resistant to water.

For $10 or less I would look at some of the very simple but reliable 2-D cell non-LED, incandescent, industrial flashlights. They put out a lot of light and usually will run for about three to four hours on a set of batteries. More than enough time to get your act together, change a tire, practice your first-aid or signal for help.

With judicious intermittent use it will serve for several days. With a single change of batteries, a week. Don't forget to get a spare bulb and keep it handy. Many industrial flashlights have a built-in holder for the spare bulb. Usually behind the reflector.

The weak point is that such a flashlight is fairly heavy and bulky. Which is okay if your carrying it in a car or around the house. A bulky flashlight can also be handy if you need to back off a dog.

If your willing to swing for a few dollars more I would get a 2-AA Maglight for about $10 and a Nite-Ize LED conversion kit for $5. Your 50% over your $10 budget but you will have a compact and reliable little flashlight that puts out a good amount of light and it will be a LED light so your not going to be burning through bulbs or batteries.