I have been led to believe that you do not sharpen the "blades" of a reel mower! You must sharpen the "bed-knife" and adjust it for the best cut. The "Bed knife" is the horizontal plate that the reel runs across, and which gives the Scissor sound effect when mowing.
You might inquire at a golf course mower shop if they could give some advice, although a lawn mower repair shop can probably do it for you.

You can't actually "sharpen" a reel mower without both a reel grinder and a bedknife sharpener. Each piece is several thousand and are made by Neary, Ideal, or Foley.

That said, you can backlap the reel to the bedknife. Requires grinding compound such as Clover, and usually a bell crank to reverse rotation of the reel. You need to gradually adjust the reel into the bedknife as you backlap, until you have contact across the full with of each blade in the reel.

Then, after cleaning up all abrasive, you need to adjust the reel relative to the bedknife so that you are just shy of contact, and check at several points on each blade, so that it is cutting like a pair of scissors.

Don't know how your mower adjusts. On professional Trimmers and Tru-Cuts, the reel rotates in eccentric bearing housings, camming in or out relative to the bedknife. On the cheaper Trimmer Homeowner, the McLane reel mowers, and the old King O'Lawns, the reel is fixed and the bedknife adjusts in and out relative to the reel.

If it sounds complicated, it is to a degree, and also very time consuming. Takes a pro an hour or more, which usually means a bill of $75 to $100.