Step 1: Get wood
I used a "pen blank", that's a 3/4 x 3/4 x 5 inch piece of wood. Woodcraft sells a five pound grab bag for $13, so I have plenty of wood.
In addition to the wood, you'll need a 5/16th drill bit and a 5/16th hardwood dowel.
You can start by drilling a 5/16ths hole down the center of your piece of wood. How deep? Less deep than the wood, or if you need exact numbers, the hole should be x-1 inches deep, where x is the length of your piece of wood. Use a drill press and a vice, if you choose to freehand it, please clamp the wood in something, you don't want splinters do you?
Step 2: The window and lip
Start by using a razor saw to make a transverse cut approx one inch from the drilled end. Cut down as squarely as possible. Stop when the blade of the saw extends approx 1/8th of an inch into the hole.
Now using a chisel (mine are muy dull), cut towards the slot you just cut, making about a thirty degree incline, make the cut as smooth as possible. Stop when the you have sliced into the bore hole about an 1/8th of an inch.
Step 3: Fipple and windway
Now what you've been waiting for, blow your whistle, sound okay? Great, if not, back out the fipple, or push it in a little farther, still bad? make another fipple, try two or three, see which sounds best. A pair of vise grips makes adjustment easy.
Once you are happy, change nothing! Use some thin super glue and wick it in around the fipple, don't get it in the wind way.
Step 4: Make it pretty
A confession, I didn't use my drill press lathe, I had such a good time using it, that after careful accounting and discussions with the wife, I went to Woodcraft and bought a Jet mini lathe. I just have to give up beer for a year.
So without further ado
I chucked the whistle into my lathe, using a scrap block to prevent the tailstock from pushing the fipple out of place.