I had the same problem. Wiggling did work for some time but after a while the Kindle did not charge at all. I took it apart and noticed that one of the 5 USB contacts was broken off. I think the contact became first loose and with the wiggling finally broke off.
I ended up getting a new micro USB port and soldering it onto the board. Really tiny contacts. You'll need to work with serious magnification to see the individual contacts.
There are plenty of youtube videos showing how to do it. With the new port, the Kindle works again :)