Been there, done that. Now that I have a bigger lawn, using a reel mower is too much like work! (I value my free time.)
Anyway, you can go to My Favorite WEB Search Engine, enter in +sharpen* +"reel mower" (quotes and '+' needed), and press Search. You'll get a number of 'hits' on the subject.
You can check out Clean Air Mowing for lots of help with Reel Mowers (the technical name for them.) There are links to a discussion forum and you can e-mail them directly. There's a 2nd Forum I found for them that I can't seem to find via their main WEB page but it's Clean Air Forum 'X'. Do search the WEB as I stated for more links.