This sounds like a battery issue giving the age of the device. Lithium Ion batteries are only made to charge about 300 to 500 times before you start to notice performance decreases. We see issues with Apple devices not working correctly even when plugged in if the battery is bad. This is likely why iTunes can’t see the device. If you never get the device to fully power up, then it’s also likely that’s the reason iTunes can’t see it. Honestly, the price of the repair might be more than the device is even worth even if you do the repair yourself. With the age of the device, finding a good battery might also be a problem.