I am not sure I understand your question correctly:
I don't understand why the remote server would "uses this encrypted ssh private key", you speak of it as it's the same encrypted private key on your desktop?? why??
If I omit this phrase (and the remote server just have your PUBLIC key in the remote login's relevant authorized_keys file), then :
- you (allegedly) forgot a passphrase of your local private key
you have an agent which starts at boot and which automatically gives you access to the key (ie, it can use it to grant access when you are connecting via ssh to the remote server, which itself probably have your public key). So the agent knows the passphrase and can decrypt your local private key.
are you sure that private key is encrypted? (you could stop the agent and try to :
ssh -i /path/to/theprivatekey remoteuser@remoteserver
if the key really is encrypted: the agent that is run at boot apparently have a way to provide its password. I don't know that agent, but maybe it's via a config file (I hope not), and maybe via some other mechanism (itself could have another encryption which holds the private passphrases? protected in some way, maybe via your own login credentials?). But : once it is loaded and using it (ie, at least when accessing the remote ssh server) it have at one point the unencrypted private ssh key in memory. At that point you can read it if you have root access to your local machine by accessing the ram (but you'll have to find how and where to look).
Hope this helps (and please clarify the obscure parts in your question or correct my assumptions!)
Edit : another options, and much simpler: that agent grants you access : use that to recreate a new pair of private/public keys, and put that new public key (in the appropriate format! depending on the type of remote sshd server!) in the remoteuser@remoteserver's authorized_keys file. in ADDITION to the current one (the one your agent is using). So that you can always use the agent, and also use the new (local)private (remote)public key pair. (read about ssh and how to create those keys and what to do with the public key (ie, put it on the places you need to log to) and private (ie: keep it SAFE, YOUR EYES ONLY, NOT WORLD READABLE even if encrypted with a passphrase).