Should you be looking for a search engine to help you find midi music files, don’t miss MusicRobot.com, which is simple, free and doesn’t require registration of any sort.
You can search for MIDIs by name of the song or by artist; you’ll then get a search results page with links to the sites holding the music, which you can then go on and download directly. Interestingly, you can use the site to search for the lyrics and guitar tabs of the music you are after as well, though note that the site searches only for terms that match the beginning of your keyword, so you might need to try several times adding and deleting particles like ‘The’ and ‘A’. Once you got your results, you can view them by length, which in turn will allow you to quickly identify if it’s a file you already have or not. While it seems clear that the site is really a project that has some years and probably not too much maintenance to it, I reckon it could be a good idea to allow some form of listening to the tune online without having to download it. 
