Change Local Admin passwords across the network
Based on the fact that Remote Management and Remote Login are disabled, you are limited to two pathways:


Installing remote screen sharing software like TeamViewer  to connect to the machine via the web which you can then administer.  The downside to this is that decent performing apps like this will cost money - and they're quite expensive
Going to each machine and not just enabling the password, but turning on remote login and/or management as well.


I prefer the latter option because it will ultimately give you much more control over your network connected nodes.  As for security, you can configure sshd on each machine to only allow logins from an SSH key.  See this answer on how to log in with keys rather than username/passwords.

You didn't mention how many Macs you have on your network and the project could be considerable. However, just chalk it up to experience as to why enabling (but securing) remote management tools is critical in networks.

You can change the password via the command line:

dscl . -passwd /Users/Lele