Sounds more like you are having issues with the box not reading games over the lack of xbox live. yes there is a fix but it is pointless unless you have another xbox's that is broken that you pulled the keyvaults from prior to it not working.
The xbox is not reading original games likely the laser has become weak look into "xbox 360 POT Tweak" to remedy the disc's not being readable. also this will not fix the problem forever as the laser is coming to the end of its life and boosting the power will reduce its remaining life span look into getting a replacement drive for your xbox as it is only a matter of time until you will need to replace it.
Drive replace is easy as when you buy a new drive you can just swap the pcb board in the new drive with the one from the old drive and you will be up and running right away ( no need to reflash the PCB with the motherboard matched keys.)
Snausage