My nieces pet rat

Question
Ok so here's the situation. My niece has a pet rat. Female about, two years old. She suddenly became sick and we're unsure if she's dying. She tries to eat and drink but ends up pushing the food away. Her back half doesn't seem to be mobile and she keeps swaying her head back and forth. Her tail is very stiff and doesn't seem to want to move. There's also a reddish discharge coming from her eyes. Her breathing doesn't seem labored at this point. I've told my niece to try to keep her clean and to keep trying to give her water and/or food every few hours. She doesn't seem to be in pain. But I'm unsure exactly how to tell.
Here's the problem. We cannot afford a vet. If we could she'd be there. What we need to figure out is if we should keep nursing her for awhile or put her down. I know you're not a vet and I read your instructions carefully and I'm sorry to directly go against them, but I'm unsure what to do. If we need to put her down is there a humane way to do it at home? If she can make it through this what are the best ways to help her along? Please, any advice would be much appreciated.
Once again, I understand you're not a vet, and I apologize. But we really cannot afford a vet. Thank you for your time.

Answer
Sorry for the delay. I had some kind of ugly flu bug and kept trying to keep up with helping everyone on here but feeling so lousy didnt help matters much.  I am sorry the little rat is sick.  I do have to say that there is NO humane way to put the rat to sleep on your own even though some people do have their own ways, if something should go wrong the rat will suffer terribly and I to be honest, I already come away from my computer more often than I like feeling bad after reading sad stories about sick rats that I cannot help at all. It sounds like the little rat has either had a stroke or something neurological going on. I would offer some baby food, see if she wants to lick it. If she should stop responding totally, there are vets out there that have a heart and may offer to put her down free of charge. I know that there are vets in my area that would do it.  Its worth a shot to be totally honest. And Trust me I know how hard it is now days to not be able to afford vet care. I would change what I have up on my instructions but some people have rats that are really suffering in sheer agony and need to find a way to get the rat to the vet even to put it out of its misery, and this is why I have it on my profile to see a vet etc...etc...   Money is tight with everyone now days and a vet is more of a luxury to many.  Hoping things get better with our economy and SOON!!  How is the little rat as of today?