One night I came home to find my Macbook (model A1811) sitting in a puddle of water thanks to my brother. I of course let it dry for a few days and at the end of this period I turned it on and was delighted to find that it worked perfectly except that even though the battery seemed to work (when I pressed the indicator button the green lights lit up to show the charge of the battery) the laptop seemed that I couldn't recognize it. I was fine with that as I could still use it plugged in. However, a few weeks later I found that when I tried to use my computer the screen was very dim and after a few seconds it would go to sleep. I could wake it up but it would just go to sleep after a few seconds again. If I turn it off and then power it back up the screen is fully bright and it gets part way through starting up and then goes to sleep. I can keep waking it up and then it gets a little farther along booting up, then it sleeps again. After doing this a few times I will get it started up, but since the screen then goes dim and it goes asleep every few seconds, my macbook is pretty much useless. From what I can tell it looks like my macbook thinks that it's closed, and that's why it just keeps going to sleep. I did some research and it seems that the battery cable is the part responsible for sensing if it is closed or not and also since I have problems with it recognizing the battery would it be safe to assume that this is probably the part busted and the one that I should replace?

definitely try the battery cable first as it is in charge of charging the battery and contains the reed switch for putting the machine to sleep.

I now opened up the machine and looked at the cable. Based on just looking at it, you can tell that it is fried, confirming that the battery cable is indeed busted. The only problem is that it looks like there is some discoloration on the logicboard right where the part plugs in. I'm hoping this is just from the battery cable and the logicboard it's self is not damaged. Would it be possible for only that one part of the logicboard to be fried and for the rest of it to still work, because as I said the rest of the computer functions correctly.

Unfortunately its very possible. Best you can do is clean it off with compressed air or very lightly with a dry rag and try a new battery cable.