Define rigid? Knife edges are supposed to be hard, but not brittle. Tough...They should not have any soft spots. By definition all steel in any condition is rigid if you ask me...
And you 'heated' to non-magnetic, not 'annealed.' Annealing is a process much like normalizing, heat treating and tempering are processes. Using the proper terminology is the only way we will be on the same page to help you.
What is your equipment setup? Forge? Oven? Torch? What kind of oil? What temp did you temper at? How did you check hardness? Etc...
Files aren't 'sort of' mystery steel, they are mystery steel. Unless it is somehow marked. Just judging by your original post, messing with mystery steel and getting a predictable result is probably beyond your skillset at the moment. Use known steel and follow a specified regimen, or have it professionally HT'd.
If it was a Nicholson, it is probably 1095. If so, non magnetic isn't hot enough. You need a shade hotter and a FAST quench. I work a lot of 1095 and found accurate temp control and quick quenching in water was the only way to get repeatable results.
If the edge didn't get hot enough, the body of the blade definitely didn't. The edge is the first part to reach temp. If you have hard and soft spots, another tempering cycle won't help.
Read the stickies... most of your answers can probably be answered there.
Cheers.