iPhone won't turn on and I am unable to put it into DFU mode

The only thing you really can do is get someone to replace the home button ribbon cable for you, sorry.

The physical home button is likely not the problem. On the other side of the iPhone screen is a sensor which detects if the button was pressed. This is the likely the problem, and you can find them on eBay as a home button as a home button ribbon cable. However, it may be that the connector terminals on headphone jack are no good and that needs repair. I'd strongly advise trying to find a pro to do this repair for you, the iPhone 5C is no fun to work on, the screen is hard to reconnect "just right". So, if you have little to no experience in iPhone repair, you'd be better off seeking a pro.