Your story hits home quite a bit actually my late fiance had transverse myelitis wish paralyzed her at 13 and caused her a whole other bucket full of health issues afterwards.
Wish I could help you monetarily but unfortunately I cannot. What I can do is offer my tech knowledge as far as a system build. I would highly suggest you build the system your self and I would be happy to walk you through any questions via skype or something simmilar once your able to build the system (unsure of your technological expertise so this may be all irreverent)
Here is a system build that can easily stream league at 1080p resolution and maxxed out everything would also be able to stream most games at 1080p and max settings some game you may have to turn a few settings down (such as shadows and Anti Aliasing). If you searching for something cheaper I can place a cheaper build that will do fine streaming just league of legends (I can actually stream on my system wich is a [email protected] with a GT610 graphics and only 4gb RAM at 480p stream quality)
Here is the build.
PCPartPicker part list: http://pcpartpicker.com/p/VpRKdC
Price breakdown by merchant: http://pcpartpicker.com/p/VpRKdC/by_merchant/
CPU: Intel Core i5-4690K 3.5GHz Quad-Core Processor ($214.95 @ SuperBiiz)
Motherboard: MSI Z97S SLI Krait Edition ATX LGA1150 Motherboard ($100.98 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($59.99 @ Newegg)
Storage: A-Data Premier Pro SP600 256GB 2.5" Solid State Drive ($99.99 @ NCIX US)
Video Card: Gigabyte GeForce GTX 960 2GB Video Card ($199.99 @ Amazon)
Case: Cooler Master Elite 430 ATX Mid Tower Case ($40.99 @ SuperBiiz)
Power Supply: Corsair CX 750W 80+ Bronze Certified Semi-Modular ATX Power Supply ($74.99 @ Micro Center)
Total: $791.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-02-06 06:46 EST-0500