one thing puzzles me - if the remote system already has debian installed, why do you want to install it again? why not just upgrade it or install/re-install packages as required?
but, ignoring that, what you want to do is quite possible. I've done it several times (including converting a couple of remote HP/Compaq machines located in the UK from RHEL to Debian, while i was here in Australia). It tends to go smoother and with less risk if you have a remote management card (like iLO etc) in the server but it is possible (just riskier) without one.
the general idea is to install debian into a spare partition (the swap partition can be used for this if there's no other free space available), chroot into that partition, install sshd, configure grub and anything else that needs to be configured (fstab, for example). you say your system has 11GB free, so you can use that.
if the existing system is debian, you can use debootstrap or cdebootstrap to install debian into your spare partition.
if the existing system is not debian, use debootstrap (or even the standard debian installer) to install debian into a subdirectory (or a xen/kvm/virtuablbox vm) on a local system and then tar it up. scp it to the existing system and untar it into the right location.
as with any major system "surgery", make a plan of what you're going to do and the order you're going to do it BEFORE you start doing any work. the very process of writing down the plan will remind you of other things you need to do. then stop and re-read your plan and make any corrections or extra notes that you need to. do that a few times, until you're sure you haven't forgotten anything.
try to design your plan to put off the "moment of no-return" to the last possible moment in time. this generally means a lot of safe, boring preparatory steps, with one last step to activate all the previous steps....and, whenever possible, leave yourself a way to revert/undo each step. for example, set up grub so that the next reboot ONLY will boot into your new environment but subsequent reboots will boot into the old environment - that way if it doesn't come back up, you can just power cycle it. if it works, then you can manually change grub's default.
if possible, practice the procedure on a local machine....with no keyboard or monitor, just as you will have with the remote machine.
at some point, though, you're going to have to gamble that you've done it right and reboot the machine. it's at this point that having a remote console is invaluable. if you don't have one, try to arrange a specific time for someone in the remote data center to be available to follow your instructions by phone/email/irc if necessary.