My kindle will not recharge because the charger plug will not stay in.
connector on kindle seems to have been
1st. You must verify that the charger is working by testing the voltage at the plug.
2nd. Using the Search at the top of this page; How to replace charging port
I had a universal charger that included many, many different sizes of plugs. One fit perfectly into a worn-out tablet saving the expensive repair costs. (and a lot of patience !)
You might look into that option.
The problem is inside the 'jack'. There is a S shaped spring that locks the plug. It is worn nearly flat and no longer 'snaps'.