Hi,
I will first say that I'm not a knife maker. But I've been a machinist for 20 years. I don't know what your flycutter looks like. So I will just show a general grind for HSS.
Like TekSec, I don't recommend using a drill press with a flycutter. The DP spindle isn't designed for side loading like a mill is. And due to the single cutting point, they tend to cause a pounding stress with loading and unloading of the tool. You could possibly find yourself picking sharp pieces of steel out of yourself.
OK, the tool failure could be caused by several things. First off, are you trying to cut annealed steel or heat treated steel? If you are working with a hardened piece, you're screwed. No matter what you try, your cutting edge will fail. A general purpose DP can't be run at a slow enough RPM to do the job without tool failure. Even in the annealed state, you might not be able to run slow enough depending on the diameter of your flycutter.
Which brings me to item number 2, Speeds and Feeds. Every material has it's own cutting speed. Most common tool steels used in knife making would have a cutting speed of between 75sfm to 90sfm (annealed) for HSS tooling, (sfm = surface feet per minute). we use a formula to figure rpms: SFM x 4/Tool Diameter. So a 2" diameter tool running at say, 80sfm would spin at 80x4/2 = 160rpm.
If you run too fast a rpm, you will work harden the part and have a failure. If you dwell too long in one spot, you will cause work hardening and cause tool failure. Some tool steels are more sensitive to work hardening than others.
Perhaps if you post some pictures of you failed tool, I might be able to tell you what happened.
dalee