WebSphere JSSL0080E: javax.net.ssl.SSLHandshakeException - The client and server could not negotiate the desired level of security.

If you see the following exception while or after starting your server (nodeagent or app server): 

JSSL0080E: javax.net.ssl.SSLHandshakeException - The client and server could not negotiate the desired level of security. 

then big chance is that there's something wrong with internal certificates between WAS instances, either:
1. Hostname has been changed and hostname verification is turned on in SSL setup. you may check this here:
http://www.ibm.com/developerworks/websphere/techjournal/0612_birk/0612_birk.html
in paragraph:

The default trust manager, IbmX509, performs fundamental certificate validation, including the certificate signature validation (ensuring it has not been modified) and certificate expiration validation (ensuring it has not expired). This trust manager does not perform hostname verification by default, although you can set the com.ibm.ssl.performURLHostNameVerification=true property in the security custom properties to enable this function for URL connections only. If this is done, the trust manager will ensure that for URL connections, the hostname specified on the connection matches the SubjectDN in the certificate returned by the server (just as Web browsers do).
(recommend reading the whole article btw.)

2. Something went horribly wrong in terms of cell synchronization, so I'd suggest stopping servers and nodeagent on remote machine and running ./syncNode.sh from profile's bin directory. not sure it will work as probably wsadmin script client won't be able to connect to SOAP port due to root cause. hence, we're back at point 1. and I guess only manual copying of trust/keystores from dmgr machine to remote machine might help.

hope this helps, or at least points to the right direction.