It is going to be difficult without having a two-way serial port.
Hard to make changes to the Kindle when you can't send it any commands.
You write of an "SD Card" - what model of Kindle is this? A K1?
Or have you modified the Kindle to add the development board's SD card socket to it?
ah, but your question - -
At this point, you need to force the Kindle's boot-up sequence to re-build the user data partition.
On an un-modified Kindle, other than a K1 model - -
That can be done by writing 16, 512 byte blocks, of zeros to /dev/mmcblk0p4 and then doing a re-boot.
The kernel's initramfs will detect the missing partition and re-build it.
But since I don't know what your device is, and what hardware mods you have done. . . .
I can't say if that would work in your case.