This is a common problem with the Kindle. If you are out of the warranty period I would first try a different cable even if the cable you are using works with other devices. If this still does not help try tilting the plug once it is plugged into the Kindle. Someti,eps wrapping the cord under the Kindle will hold the plug at enough of an angle to maintain contact and thereby charge the device. Not ideal but effective and ver cost effective.

The socket it engages is likely broken loose inside the unit, get someone who knows how to safely take it apart, if under warranty this may not be covered or if unit opened the warranty may be voided.

Not in most cases something you if a do it yourself person should attempt, most laptop repair places should be able to take care of this.