Can you give some examples of what situations make your ankles feel week?
How is regular walking?
One of the best things in helping increase strength and stability in my ankles, was to walk uphill (treadmill is great for this).
Do NOT hold on to the railings, handlebars, or any part of the treadmill. These do not mimic walking in "real life" and do not strengthen your legs much - especially when it comes to balance and stability.
Go as slow as you need to, with a as little of an incline. Slowly increase and test new levels out (you can do short 30-60 second intervals). It's SO much better to go snail slow, and walk with a decreased incline (or even flat), than prematurely increase either and hold on to the treadmill. Forget the awful form - people who do that aren't even getting much of a workout, since it dramatically decreases the effort on your legs (largest muscle groups, and calorie burners in the body).
But anyway, start walking. If you want to slowly build strength, slowly add incline.