How does Judaism address this problem then?
I have not seen this question brought anywhere in Jewish sources. Mostly because I haven't studied Jewish philosophy as its not an encouraged area of study. So I can only derive an answer from other sources.
According to the Talmud, evil events are a perception. We are encouraged to constantly recognize that everything that happens to us is for the best, even if we are not able to perceive what might be good about it. Suffering, is understood to be a positive thing - one reserved to purify those that G-d wants close to Him. Were we able to see the benefit, we'd not want it to be taken away from us in most cases.
The existence of the perception, is a development in the progression away from G-d. If the greatest pleasure is the return of the person to his Source in G-d, then the opposite direction is increasingly greater suffering. Without this continuum, there would be only stagnation.
This directly relates to what is actually evil - movement away from G-d's Will: If the ultimate good is defined as G-d, then anywhere you stand on the continuum between G-d and the opposite end, in front of you will be good and behind you will be evil (assuming you're travelling in the right direction). Depending on the person's movement, what is good or evil for him will constantly be changing. In that respect, the Creation (as being increasingly further from G-d) is evil.
What we have, is sort of a split between the choices a person makes and the events that occurs to him. From my perspective, I can choose to do evil. But from the perspective of the person that I am being evil to, I am benefiting him (whether he ultimately recognizes that or not). Because of this, there's no contradiction to G-d's control of events: we can choose to go against G-d's Will (ie. to transgress what he expects of us), but the outcome of our actions will only be according to G-d's Will (it will benefit the victim).
The reason why we are required to go through this world of testing, is not for G-d, but for us. G-d already of course knows everything we might choose. Here the idea we have is called "bread of embarrassment". The idea is that a person who receives charity, doesn't appreciate and enjoy the money he received as when he works hard for it himself. Consequently, the existence of the concept and our existence here, causes us to appreciate and enjoy the reward we will receive for passing our tests, to a greater extent. I guess its kind of like getting more bang for your buck. Which is consistent with the idea that G-d wants us to get the greatest possible enjoyment.
In summary. There's no reason for G-d to have created the creation any other way that He did. Its a logical progression and is ultimately beneficial. Anything that is perceived of as bad, is a result of faulty perception, not bad design.
