There is a cheap easy way to sharpen a reel push mower.
Buy valve lapping compound at the auto maintenance store like AutoZone ($5). Then adjust the mower's plate (the cutter bar) at the bottom of the reel mower to where it makes contact with the blades throughout the blades' entire rotation. Then put valve lapping compound on every blade using your finger, rotating it as you go along to get all the blades. And then on a smooth surface without cutting anything, push the mower until all the valve lapping compound is gone, keep doing that until it is sharp.