Since the question is resolved, I cannot add a proper answer, but here is a more in-depth answer:
1) "Headache" can "go away" (on it's own, or because somebody did something to make it go away) so "go away" is the verb that can actually describe the action of the "headache" itself. When it comes to "headache" (as opposed to "wound", for example) it cannot "heal" itself. I think this is because there is no way for the healing process can be observed, and healing is a process that usually requires some external assessment. Only an external force can "cure" "headaches" and since "cure" as a verb usually implies a long process, it would not be applicable to a single instance of a "headache" (which usually lasts several hours or so). But somebody can cure "headaches" (note the plural) - as in chronic or frequent headaches. Example: "Jane suffered from regular headaches since childhood. Now, at last, the famous doctor was able to cure her headaches."
Finally, only the person who suffers from headaches can recover *from* a headache. Recovery is the process of returning to previous/normal state. Headache's normal state is to hurt :) Therefore if the "headache" itself were to "recover", it would mean that the headache was back in full force - quite the opposite of desired effect :). So a person can recover from a headache, especially if it's a very strong headache that incapacitated a person for a while or caused some serious damage (like hallucinations, for example). An example usage would be: "John was so tired after 10 hours of constant pain that it took me 4 hours of sleep to *recover from* his headache". Does this make sense or is it too confusing?