Pretty sure we had this happen for a friend when I bought him a Kinect for his birthday his console was a older white original Xbox 360 not a slim version.  I am pretty sure though we just managed to reinstall the update using a new disk rented from a video shop.  This I have to add was on a console which was modded.

I am thinking it's more likely needing a new dvd drive.

I also had a issue similar to this when doing a recent update myself would have been the update prior to the one you've had the issue with it started the update then just came up with the error update failed please try again.  Problem was that it didn't take us back to the dashboard it just sat there with no choice other than to power off the console, which upon restart would go straight back to the same error.

I ended up with a near identical conversation with MS about where to from here and they were apologetic and advised sending in for a repair also.  In the end I took it to someone I knew who mods consoles got the latest mod done and got the console fully working again with no issues since.  All up cost just under $100.

At the end of the day it's fine to mod your console and you should have no online issues if you don't play backups at all. Of course making and or playing of backups is illegal and breaches MS terms of use so can and will eventually lead to your console being banned from Xbox live which is not great.  Plus if going down the modded path you risk the chance of future updates potentially bricking your machine (or at least forcing you to remod the console).

Either way I hope it all works out for you in the end and that you get your issue resolved without having to spend much more $$$.  :-)