If the Micro USB charge socket is loose, then very likely it is broken inside. I'm afraid the solution is to get a new Fire tablet. They are so inexpensive that it is uneconomic to fix the broken Micro USB charge socket. Plus they are not designed for disassembly and repair.
I have looked to do a reset but as the Kindle will not charge at all I cannot do that. The charge connection is quite loose so I fear that may be the problem. Is there any suggested solution?
I have had a "loose connection" with micro-USB connections. Invariably, for me it has been the plug on the charging cable that was causing the problem and not the socket on the device. It was easy to test the device with another cable, or even buy a generic replacement cable before scrapping the device or trying to get the socket replaced.
I suspect that the industry might by design make the socket stronger than the plug, so that any wear and tear will preferentially damage the plug on the easily-replaced cable and not the socket on the device.
Only once did I need to change the micro-usb socket - it was on a phone, I caught the cable as I stood up from my deask. The phone was plugged in and fell to the floor. There had clearly been a stong jerk on the plug and the socket would no longer hold any plug securely. Luckily, for that phone I was able to get a new socket on ebay from China and repair it myself.