The charging port on the Kindle Fires are a notorious issue. It starts off like you're experience, where you have to wiggle the cord. Soon even that doesn't work.

If you aren't comfortable soldering, you can try this... insert the cord into the charging port, and apply pressure to the side of the cable close to the charging port. If you had the Kindle laying on a counter, apply pressure diagonally down and to the left/right. Hold it here for a couple of minutes and see if the charging status screen appears. If it doesn't try diagonally up and left/right. Typically, you will find the direction it likes if a few minutes. After you find the direction, you can fully charge it without having to apply hand pressure for a couple of hours, by standing the kindle on top of the charging cord/port - think of the cord as a very flimsy single leg - then prop the kindle against the wall so it keeps standing on the port. You may have to manipulate the cord direction, to recreate the same diagonal pressure. This is a work around to get the Kindle Fire to charge. As stupid as this sounds, it is what a friend of mine does with his Kindle Fire, and he has been perfectly content to charge it this way for about a year now.

What has happened is there are cracks in the solder joints of the charging port, so you have to physically manipulate the charging cord to provide the port with enough contact to charge. The long term fix is to open the Kindle and resolder the charging port. Here is a video of repairing the charging port: https://www.youtube.com/watch?v=dxqq_XJA...