Is there any way that makes the new SSH Key work with an old Droplet which is built with another SSH Key? It seems that every time I update my SSH Key, I need to build a new Droplet to use the new SSH Key.
Since you are referring to the ssh keys being put in place during Droplet creation I assume you are talking about adding an additional/new ssh keys to the root account?
Assuming that you still have access to the running Droplet you can yourself add an additional ssh key by editing /root/.ssh/authorized_keys. The more generic answer, for any user, is that a user's ssh key(s) are looked for in ~/.ssh/authorized_keys.
If you on the other hand have looked yourself out of the Droplet you ought be able to reset the root password by way of the https://cloud.digitalocean.com/ control panel.