I think you found the best answer yourself: use a predictor! There are several out there...
You suggest that there should be a Secondary Structure Database. I'm not sure that makes much sense, let me explain my point of view (which may not be that of everyone): most often, the data that is found in databases is the "state of knowledge" of the described object, based on experimentation.
That may be the case for secondary structures of proteins, but only in the case where the said proteins have been crystalized. In those cases, it is not only the secondary structures but also the tertiary structures (with the caveat that the crystal structure of a protein does not prove "all" states that a protein may take in real "dynamic" physiological conditions).
For all those proteins that have not been crystalized, then we can only rely on predictions. And I use them quite frequently: they are extremely useful! But as far as I know, no prediction is accepted as fact. They're "educated guesses" that are often correct, but sometimes wrong. The results may differ from one prediction method to another. Also they change each time the algorithms are improved...
If there was a database of predicted secondary structures, people would likely take them for granted (make the equation prediction = fact) which would be quite "unscientific".
I think such a resource would be more of a hindrance than an asset to the scientific community...
How can I download dssp file for large scale of pdb or I can download complete file?