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How long have you owned the Kindle for please?
Please let me know when you are able to see my reply.
Since this is an older model Kindle, and what you're describing is a hardware failure within the unit. Since you've tried 3 different charging cables with the same results, and have to lay the Kindle down a certain way in order to get it to charge, this would mean a cracked solder joint on the USB charge port. There's no pushbutton fixes for this and if you want to repair this yourself, you will need to heat up the solder pads on the circuit board by dismantling the Kindle and then use a heat gun on the High setting for 30-45 seconds.
However if this is not something you're comfortable with then you would need to contact a 3rd party Kindle repair service such as Mission Repair at the link below. They generally charge $79 for this type of repair:
Other than that, I'm sorry for the bad news and wish there were a quick fix.
If you need further assistance let me know. Please do not forget to click the 5 STAR RATING, at the top of the page for my efforts. It is my livelihood and you can follow up at any time if needed.
Do you need further assistance with this?If not, could you please click the 5 STAR RATING, at the top of the page for my efforts? It is my livelihood and you can follow up at any time if needed.