To be honest, the proper way to do this would be via VLANs and a managed switch environment, with a router (or switch) doing the inter-VLAN routing, so you put the call recording in the voice VLAN, and then devices on the data VLAN can still talk to devices on the voice VLAN, but their DHCP scope (and broadcast domains) are unique and won't interfere with each other.
This is a very common implementation with VoIP networks, having two distinct networks and two NICs on the various devices that need to talk to both networks is usually reserved for very unusual circumstances, typically where security protocols prohibit network interconnection for some reason (and these reason are usually unfounded, but occasionally are valid).
And Tech Electronics is correct about the Mitel 5000's DHCP server, by default it will only respond to phones requesting DHCP, but this can be changed to allow it to hand out IP information for other devices as well. I do know on some of the original software releases that had DHCP server embedded in the 5000, I had some issues getting it to work, especially if this was a system upgraded from a version of software which did not have DHCP server previously. Typically stopping the DHCP server, resetting the DHCP server to defaults, setting up the DHCP scope again manually, and then starting the service usually fixed it.
If you are not familiar with router programming, switch programming, and/or VLAN implementation, this is not a DIY project and you should seek professional assistance.