You could have some reflexive pain after icing as the blood flow returns to the area. I would however not ice for more than 10 minutes at a time. You can then let the ankle return to body temperature after which you can repeat for another 10 mins. 5 times a day is a little excessive in light of the fact that you cannot remember injuring it.

There are other things that could cause lateral ankle pain: referred nerve pain, local inflammatory conditions in the joint etc. If you are in persistent pain then perhaps another review (maybe with another therapist) might assist to consider an alternate diagnosis. Sometimes a clinical assessment indicates one probably cause but subsequent treatment suggests perhaps it is something else. All part of a normal process of elimination.

If all that fails you could scan the ankle to see if that shows anything further. But mostly that should not be necessary. Do let us know how you get on.