Well, I suppose it is possible that she got this entrenched in her mind since childhood and finds accepting her two legs as impossible...

There's no way I wouldn't call this a mental illness... but the possibility remains that she developed the insane desire and maybe the best way to fix it is to take them off... Ultimately, I think a person's body is in their own domain, so long as the action in question only effects them.

I say maybe, because I have no idea about how effective psychological or psychiatric treatments may be and whether or not some other underlying mental illness, aside from what they call BIID, prevents her from reasonably changing her mind.

Could nerves in the leg, or even those correlated in the brain, be messed up and she actually has some reason to think something is wrong with her legs?

I think BIID will stay taboo until people get together and bring it out. A hundred years ago, it was taboo to be gay in many societies, and 50 years ago the idea of transsexuals was abhorrent to most. I have tried to make the condition more understood but it is difficult to get a case out in the open by yourself.
The real issue isn't how accepted it is as being normal, but more about how accepted the idea is that she can't reasonably change -- I don't mean actually using reason but how likely she would be to ever change -- her mind about her legs.