I used to work at a golf course, they did a lot of sharpening... to
test the blades they would literally cut strips of paper with it, so it
was really sharp.

The theory is simply, you run the reel backwards, painting "lap" fluid
on the blades as you do it. The lap fluid, short for lappidary(sp??)
fluid is just a grit suspended in a fluid. the lap fluid gets between
the blade and the bed knife as it spins, and you eventually get a super
fine fit, and the blades literally cut like a pair of scissors.

How to do it - get the lap fluid online or a local pro lawn power tool
dealer. Get an old washing machine motor, mount on a board or
something, put a pulley on it, then hook the motor onto the pulley on
the blade (disconnect from motor) and let it run backwards, painting
the lap fluid back and forth across the blades and evetually you will
hear when the blade is perfectly seated, the tones change, this seemed
to often take 20 minutes or longer. As I said, then test by shearing
shear a stiff strip of paper with each blade (not newspaper), the
left, middle and right side of the bedknife. Before doing this you
should check for nicks and gouges, which you would remediate with a
file. And bedknives have a series of screws accross teh bottom, which
you may have to play with to get tension just right.

That of course would be for a major job, you can do the same with a
rachet on the bolt on the end of the blade, turning by hand as you lap
the blade in, instead of the motor.