Emulate a USB downstream port
If it doesn't work with a charger it sounds like you'd need implement a USB host as RedGrittyBrick suggested in a comment. One of the cheapest ways I can think of to get a USB host device is to use a device such as the FTDI VNC2 - Vinculum-II Programmable USB 2.0 Host. They are currently under $5 in one-off quantities but will require power, a PCB, additional support circuitry and a good deal of design and programming work.

A Linux SBC (single-board computer) such as a Raspberry Pi or BeagleBone may be a more approachable option but they tend to cost more and will also require external power. It may also require some programming depending on what support the Kindle has for Linux. Personally I'd concentrate on trying to repair the broken button.