How do I deauthorize non-existent computers?

My 8 year old laptop finally died on me. So I got a new Windows 8 machine and I'd like to back up my iPhone. The problem is, it says I have 5 computers authorized already and I can't authorize anymore. I'm not sure what other computers I authorized other than mine, but apparently I did at some point. At any rate, my old computer is dead. There's no way to revive it. If I authorized any other computers -- pretty sure I didn't -- and I can't authorize my new one, iTunes says it will delete my music and apps from my iPhone. I've been using iPhones since iPhone 3G and I'm a loyal customer. Surely, there is some way to remotely deauthorize all those other computers so I don't lose my purchases... right?