Error 4013

Ipad stopped turning on and continuously shows the need to plug it in with an apple support restore address at the top. When I try plugging it in and updating or restoring, it gives me a 4013 error message. The ipad was just months out of warranty when this happened (just over a year from purchase) and now apple were initially helpful via chat support, but I can no longer get into chat support. It gives a message about how it is no longer in warranty. Coming from a household with MULTIPLE apple products I have had repeated problems with ipads not turning on within two years of purchase. Apple tells me it is almost the price of a new ipad to fix the problem which seems insane!! Looking for ideas or support. Anyone had similar issues? I am very unsatisfied with the help I've received from apple. Certainly won't be recommending ipads to anyone I know, nor the help service surrounding this. Very disappointed customer!