I am using a router with internet access with the default gateway of 192.168.1.1 and another router with intranet access with the default gateway of 192.168.105.4 and the internet router is not being accessed even with the correct address. It thinks those two are the same.
Based on my limited understanding, a router wouldn't have a default gateway... they each have an address, and it can be used as your client system's default gateway.
What is your network mask on your client? What is the ip of your client? My guess is that your client is 192.168.105.x with a subnet mask of 255.255.255.0 based on your description. The client will send all requests outside of the local network to whichever is set as its default gateway, which should (I believe) be on the same subnet.
Since you have 2 gateways, you will probably need to do some sort of routing beyond what is provided with a default gateway.
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