Sounds like there were several issues that caused the sprain. I agree with advice given by everyone else...sounds like some neuromuscular issues (brain talk with muscle) could be the underlying cause of the 'weak ankle" as well as the fact that most of us could use balance/stability improvement. Throw in a basketful of laundry and not being able to see the last step...yep. Bad news. Could happen to many of us.
I like the idea that you were trying to be all healthy and take the stairs but just keep in mind that strength training is usually most successful when we are mindful of what we are doing...and can SEE over the laundry basket.
I'm only teasing because I so could do this myself. I have a history of clumbsiness inside and outside the gym.
LET THE SWELLING GO DOWN before you do anything else. Completely. Little puffy..nix the idea of doing more work. Let it rest first just as Fitness suggested.
When you are ready, the balancing suggestions are great. Also...consider some of the standing poses in yoga. These are great foot/ankle stability builders. Tree pose comes to mind but really any of the standing poses will do. Check out Yogajournal.com for suggestions if you are unsure where to begin or see if you can get in on a class through your university somewhere. Maybe you could take a class as an elective. (wish I would have taken advantage of that back in the day...
) Best of luck. Sorry about the twist in your ankle. Those really hurt...