Speaking as someone who has replaced a laptop motherboard before, you're better off buying a new laptop. The new part will be $150-200 to pick up, and then you've got the headache of actually replacing the thing. This involves completely disassembling the laptop and then re-assembling it.
I've done it before, but it was incredibly frustrating and time consuming. It was only worth my time because I had two identical laptops. Neither worked properly, and I had to marry parts together to get one working laptop out of the two. This allowed me to figure out how the thing was put together by taking one apart first, and then using that knowledge to carefully disassemble the other one and replace the things that needed replacing.
All laptops are put together differently. Finding a guide for how to disassemble that specific laptop will be very difficult. The best suggestion I can give for finding a how-to for it is to search for a factory repair manual for it. If you can't find one, then you could try your hand at taking the laptop apart with no instructions and figuring out how to put it back together yourself.
Wish I had better news for you, but that's the state of things. Honestly, you'd probably be able to get a refurbished or used laptop somewhere for only a little more than this one would cost to fix.
If you're really dead set on finding a replacement motherboard, search "gateway nv5207u motherboard". Did that myself and came up with lots of results. As far as exactly which you need, I can't really help there. It looks like that model only has one motherboard design, though.