Need help with accessing internet and another LAN with 2 NIC computer

Hello. I'm finding this very difficult and frustrating at the same time. I have a LAN, and in this network there is a computer offering web services on port 8443 for example. Now, this LAN isn't connected on internet and the ip of the computer offering the services is something like lets say 10.31.0.14. If I connect my computer on this LAN I use the IP 10.31.0.44 (on one of my NICs) to access those web services. Now the tricky part. I need to also connect my computer to internet via the other NIC on my motherboard. This way I can remote on my computer via Internet and use those web services in the no-internet LAN (10.31.0.xxx)
Now, I have tried bridging the connections together but I get "cable disconnected" on the LAN NIC. If I remove the network bridge everything comes back to normal, but with both cables into my computer I can only access the internal LAN (with no network bridge). If I make the network bridge then I can connect to internet but not the LAN (the LAN NIC says no cable connected even thou there is a cable connected). The NIC with the internet connection has DHCP and it gets and address like 192.168.3.xxx. What can I do so I have my computer connected to the internet, as well as at the other LAN (that doesn't and shouldn't have internet) ? I am only interested in the web services offered by one of the computers in that network (10.31.0.xxx).
I presume it is possible if I have 2 NICs on my motherboard, shouldn't be difficult for one to access the internet and the other to access the other LAN at the same time. Basically I want to access http://www.example.com and https://10.31.0.14:8443 from a browser.

I use Windows 7 Home Premium 64 bit and my motherboard has 2 identical NICs.

As I understand your post:

Your Internet NIC is DHCP, which means that everything, including Default Gateway is assigned. This should work just fine.

On you LAN NIC, you should have a fixed IP and Subnet Mask, but leave the Default Gateway entry blank. That should work.

As long as everything you connect to is on the same subnet you shouldn't need any static routes, but you won't be able to use an internal domain name, just IP addresses.

I left out the gateway from the LAN nic but still no luck. I tried alot of combinations but nothing works. Either I "see" the internet either I "see" the LAN, but never both. I have selected from the bridge menu both connections and each one sepparate, nothing good happened. Either I see only one network either none either one of the connections say "No network cable attached" even thou as soon as I delete de bridge it goes back to being connected. I really need to make this happen, I also tried this on Windows Vista with another computer and still the same problem. I attached some screenshots with config and ping results. Did anyone actually tried this on Windows and worked? Because I tried this thing on 2 different computers with 2 different windows versions but the result was identical on both. Weird behavior saying network cable unplugged. And if so, then why this option in Windows? What good is it?

I added some extra info. Local Area Connection is the internal LAN connection and Local Area Connection 2 is the internet connection.

anyone? I haven't solved this problem

I'd start by removing that bridge you created and then confirm you can still ping something on your LAN and Internet.

If both are ok then next would be to check the network bindings.

Goto: Network Connections.
Press Alt to bring up the menu
Goto: "Advanced -> Advanced Settings"
Under Connections set your Internet NIC (Lan2) to be at the top.

A reboot is not required but wouldn't hurt.