My refurbished laptop broke after less than a year and I can't dispute my warranty

[Mary2451]

I bought a refurbished HP back in August or September and two days ago (10 months in) the backlights went out. I am furious. I have tried all the troubleshooting with HP (eventually called them) and took it to a local repair store to see if they would be able to fix it at a lower cost than HP. Both are telling me it's going to cost even more than what I paid for the computer in the first place. The kicker is, HP says my warranty is expired, so they won't help me with anything even though it is less than a year old. I have tried to dispute the warranty, as I believe it is ridiculous that they expect me to practically buy a whole new computer less than a year out from them giving me one that they "refurbished". BUT the link to dispute the warranty won't work. Someone please help. I cannot afford to buy a new computer every 10 months. 

(I am not trying to sound entitled, I am truly just at a loss as my computers have always lasted me at least four years and this relatively new one was a HUGE purchase for me. Quite frankly I am in grad school and this is a devastating blow educationally, vocationally, and financially. Please help.)