Thanks kenchy. That was enough of a hint that I can now ssh out of the kindle from the command line. Brief instructions for anyone who wants to achieve this:
1/ Install dropbear on a different computer, I used Debian
On the computer:
2/ run "dropbearkey -t rsa -f /tmp/id_rsa"
3/ Copy the public key (echoed to the command line after step 2) to the authorized_keys file on the computer you want access to
4/ Change the very last part of the key to root@kindle
5/ scp /tmp/id_rsa root@kindle-ip:/mnt/us/id_rsa
On the Kindle: "dbclient -i /mnt/us/id_rsa root@server" will now open a shell on the server