Windows server 2012 Domain Controller two NICs with different network profiles

You would set the default gateway on the public NIC. This will make all traffic that you don't have a specific route defined for use it.

Then, you need to add a static route to the internal network. You can do this with a route command.

route add 10.5.1.0 mask 255.255.255.0 x.x.x.x if y -p

Where x.x.x.x is your gateway on the internal network and y is the interface index of your internal NIC prefixed with 0x. You can find the index of each interface by running route print. If it was interface 2, for example, you would substitute 0x2 for y in the above example.

The -p at the end makes it persistent, so that the route will remain across reboots. If you just want to test, you can make add the route without -p and it will go away on reboot if it causes issues.

You need to read the post "How does Subnetting work". Your two NICs, despite having very different octets are actually in the same subnet because of the very large subnet mask you've used. Because your default gateway is reachable by either subnet, the system is alternative NICs attempting to reach the destination when the first fails.

10.1.8.1/8 and 10.185.1.44/8 are in the same subnet. If you want to keep this simple, just use a different RFC1918 space like 172.16.0.0/12 or 192.168.0.0/16 networks.