Do you have SIP license (assuming they are required) on the Mitel Box?
You just need a simple trunk configuration on both sides and the appropriate dialplan…I recently did this with a Toshiba PBX. I recall doing it through analog lines in the past as well.
A simple trunk configuration on the asterisk side with basic parameters like this:
That and a similar basic, simple SIP trunk on the Mitel side should allow the SIP traffic to navigate between the boxes. Ensuring the diaplan is appropriate for the systems (varies based on full PBX integration, only use for a conference bridge, voicemail system etc.)
Ensure the dialplan ‘catch’ on the asterisk side and Mitel side are also setup to process inbound calls.