They're also very close relatives to the cockroach, who can survive an extremely long time without their head and will only generally die due to dehydration/starvation.
Insects do not have a centralized brain or nervous system. All of that stuff is dispersed throughout their body. So while the mantis won't function fully without it's head (due to losing all of the sensory organs in there), they'll still fare quite well and are, by all definitions, still 'alive' and breathing.
You might actually be surprised to know that an immaculately severed mantis head can also 'live' for a couple hours, and will wiggle the antennae and behave as an insect's head head should, until it runs out of steam.
But unless the insect is fully in tact, it'll never behave as a fully functioning entity in terms of it's response to stimuli. Your mantis, without her head, can probably live for a week or so in the best conditions... But I dunno, man. I'd just feed it to the other girl. Let it serve a purpose :'| <3