When the witness does not exist and the primary goes down, I believe mirroring will be broken and you have to manually make the mirror DB the primary.

I know when we DR tested our mirroring without a witness, we had to manually make the mirror the primary. But the primary server wasn't actually down so much as stopped. Then we were able to switch back to the primary (again manually) when we stopped the mirroring on the secondary server. So mirroring wasn't officially broken in our case. It was still a manual failover, though.