Welcome to the forum.
As you know, the reel mower has a stationary blade and multiple (4 or 5) rotating reel blades.
This link has a decent sharpening process.
A reel type mower is sharpened by first ensuring the reel to stationary gap is correctly adjusted on both ends.
At each end there are one or two adjustments to ensure the stationary blade gap is correct.
The mower drive is disassembled to permit attachment of a manual drive crank handle.
Lapping compound is applied to each rotating reel blade. The reel is manually rotated until each blade has a nice uniform surface finish. The mower is then reassembled for normal usage.