What kind of ankle brace would you recommend for an inner high ankle sprain?

Lacer: Best brace is a lace up.

Depends on severity: It depends on type and severity of the injury. Have it evaluated and have the appropriate brace prescribed that best suits your condition.

More rigid: If you have truly have a high ankle sprain, then more rigid braces that come up higher on the ankle do a better job of protecting the syndesmotic ligaments. These types of braces are often provided by sports medicine and orthopedic doctors in their offices. The over-the-counter braces are fine for many of the "garden variety"/low-grade lateral sprains.