I sometimes use my brother's computer and simply close all of his open tabs, and he always thanks me.
You say you're not anxious... But I'd think more about that angle. What you describe sounds like a form of anxiety. You can be anxious and still be able to sleep.
Information doesn't have to be painful or exhausting. Why do you feel like this information is a burden? Why do you hesitate to declare a jubilee and just clear your Instapaper?
How do you relax? What are your habits like?
Maybe you're focusing your problem-solving energy on fixing this info overload problem, when what's actually bothering you is something missing in the other parts of your life?
I know what you're talking about, but I also quite clearly see how I've previously rushed to blame info overload. Actually, my ability to let info just be what it is—without it causing anxiety or unhappiness—seems to depend crucially on other factors.