I really don't think that a UV will help your issue.. I think it may be more of a band-aid for your problem. Stress is the main cause of Ick. Something is stressing your fish out.. weather it be the types of fish you have in the tank together, water quality, where the tank is in your home in relation to what is out side the tank, kids banging on the glass... something is stressing your fish.
I did a lot of research on UV sterilization and what I found is that they can be more harm than good by killing off the good bacteria in your tank that it needs to survive.. thus creating a larger problem. I'm not saying that everyone has the same results, but IMO if you can go without one, your tank health will be better.
On the opposite side, they do kill off the bad bacteria as well.. but if you maintain your tank and are careful about what you put in the tank, then you can avoid that all together.