Become a MacRumors Supporter for $50/year with no ads, ability to filter front page stories, and private forums.
You probably already know this, but in case you don't I would suggest you go here, and especially to this section for info in regards to starting the process of filing for disability. I realize I'm now way, way off topic for this forum, but you did mention that is one of your other concerns, so I thought I'd throw it out there. If you pursue it, you really will need help filling out the paperwork. Your brother sounds too busy, so maybe your nurse? Be sure to make copies of your application when it's completed, to save time if you should have to fill out other forms with similar questions or re-file.

Unfortunately, the process seems to be long and drawn out in most cases. There are lawyers who specialize in this type of filing, so you may just want to go that route. Or like I'm doing, go it yourself at first, then call a lawyer if you need to appeal the decision...but based on your posts you will want to get as much help as you can, as soon as possible. Check with your hospital/Dr.'s office to see if they have some sort of Patient's Advocate or other type of assistance programs that might help you, not only with this, but with anything else that may come up as your mind begins to adjust to your new reality.
 
Back to the point...

There are a couple of options.
First, the OP should get all of the files on the machine backed up if he hasn't already done so.

If the machine is usable, the easiest way to do this is to purchase an external hard drive and then using Carbon Copy Cloner or SuperDuper software (both are available through www.versiontracker.com) clone the internal hard drive to the external.

I'm a little confused whether the machine is functioning or not, however if it boots, but the screen is not viewable, then it could be used with an external LCD monitor. See if you can buy, rent or borrow a monitor. You will need an adapter cable that came with your Mac, or get a new one from an Apple dealer.

The alternative is to hook this Mac up to another Mac with a Firewire cable, and put this Mac into Firewire Target Disk Mode (reboot the machine and hold down the T key through startup). Then the hard drive of this machine will show up on the second Mac as if it were a Firewire hard drive, and the second Mac can copy the files off for backup. Again, having an external hard drive would be the best place to back up the files to.

I don't see any issue with Apple dealing with another person. I do it all the time for customers. Just have your helper say "I have this machine serial number #### and I need to arrange a mail-in repair or courier in repair." There doesn't need to be any conversation about the identity of the person or the medical situation, the speaker can just say "I'm not mobile, its not possible for me to bring it in.". Apple offers a service where they send a courier box to you for warranty repairs. Don't know about out of warranty.

Or even simpler deal with an independent Apple dealer... in Salt Lake City there are several, to search Authorized Repair centres go to
http://www.apple.com/buy/locator/service/ and enter Salt Lake City, UT

Or enlist the help of someone from one of the Mac users groups in your area
http://utahapple.org/
http://mac.byu.edu/
http://umug.com/
 
Register on MacRumors! This sidebar will go away, and you'll see fewer ads.