While most might assume you need a knife, blade sharpener, or other instruments to sharpen the blades on your reel mower, this is not the case. You will need three items, a crank or object to turn the blades, a lapping compound, an abrasive suspended in a thick goo, and something to apply the lapping compound with, like a paintbrush or other instrument.
To start, you want to adjust the mower blades so that they are as close to coming into contact with the bed knife as possible without actually touching it. Then, apply the lapping compound to the very edge of the blades. After you apply, you can use a crank or other instrument to move the wheels backward and continue applying until each blade has some lapping compound. Now, crank for about ten minutes or so, still backward. After a while, your arm may tire, so it might be a good idea to have another person who can take over cranking duties. After a while, you should start seeing the shiny edges appear with a bit of corrosion. To check the progress of your blades, remove the compound. If they aren't totally shiny, apply more compound and keep going. Once you think your mower is sharp, test it with a piece of paper, it should cut like scissors.