That means that the last ounce of patience she had left is now gone. Therefore it can be tranlated as both lost patience and ran out of patience.

The word иссякнуть is used a lot with water.

Вода в колодце иссякла - Means that there is no more water in the well. It's either all vaporized or soaked into the ground. All in all, иссякнуть conveys quite a long, gradual proccess. Much like "die out" I would assume.