There is no headache strong enough, that a good coffee won't relieve
I think the main source of awkwardness is the missing object for "relieve". But it's grammatically correct. Compare the (simplified) negation, which is more obviously correct:
There is a headache that a coffee will not relieve.
I would prefer
There is no headache so strong that a good coffee won't relieve it.
It's the sort of thing that many native speakers might say, but some might prefer to end the sentence with it.