German has a feature, that only a very small number of languages have (English doesn't have it): Separable verbs. »Einschlafen« (to fall asleep) is such a separable verb. It is derived from »schlafen« (to sleep), and literally is to in-sleep.
Separable verbs will be split into its parts in many grammatical situations. If this happens, the former prefix floats to the end of the sentence. (Consult your grammar book for more details.)
I compare here »schlafen« (to sleep) and »einschlafen« (to fall asleep):
Ich werde schlafen. Ich werde einschlafen. Ich werde mit dem Messer einschlafen.
I will sleep. I will fall asleep. I will fall asleep with the knife.
Ich schlafe. Ich schlafe ein. Ich schlafe mit dem Messer ein.
I sleep. I fall asleep. I fall asleep with the knife.
Ich habe geschlafen. Ich bin eingeschlafen. Ich bin mit dem Messer eingeschlafen.
I was sleeping. I was falling asleep. I was falling asleep with the knife.
So, to answer your question: Yes, »ein« can be removed, and it would still be grammatically correct. But it would mean something different.
Ich schlafe mit einem Messer ein.
I fall asleep with a knife.
Ich schlafe mit einem Messer.
I sleep with a knife.
Addendum:
(Reaction to other answers who claimed, that »Ich schlafe mit einem Messer« would mean »I have sex with a knife«. This is nonsense.)
»Mit jemandem schlafen« means: To have sex with somebody. But »mit etwas schlafen« just means: To sleep while lying next to something, or holding something:
Die dreijährige Lisa schläft mit ihrem Teddy.
This doesn't mean, that the little girl has sex with her teddybear. It just means, that she is holding her bear while sleeping, or that the bear is laying in her bed while she sleeps. So
Ich schlafe mit einem Messer.
Does not mean, that you have sex with a knife, because a knife is not a person. It just means: you are sleeping while you have a knife in your bed. (Maybe under the pillow?)