Improving your ankle strength may not be what you need. Reoccurring ankle sprains often have less to do with lack of strength around the ankle as much as a loss of balance and proprioception the ability to know where you joint is in space.
Here's how proprioception works:
Close your eyes and make a fist. Place one finger up, then two. You can sense where your fingers are, right? That's not because you're looking at them, but because you can feel them due to proprioceptors in your ligaments.
When you first sprain your ankle, you damage the ligaments, which in turn damage the proprioceptors in that area. When you suffer another sprain, it isn't necessarily a lack of strength that's at fault, but since you weren't looking at your ankle, you had less of a sense of where it was in space, so you turned it.
Re-establishing balance and proprioception will help you avoid future ankle sprains. Try balance activities that challenge your vision, like the progression below. Do it someplace where you can easily touch to regain your balance if needed, such as a doorway.
1. Single-leg standing: 30 seconds
2. Single-leg standing with eyes closed: 30 seconds
3. Unstable single-leg standing: 30 seconds
4. Unstable single-leg standing, eyes closed: 30 seconds
5. Single leg standing, eyes open turn head to the left, then right, then up, then down. This is one rep. Repeat 5 times.