I have been looking a lot on Google on how to install the service in command line (so without manual interaction) but I am stuck on how to get the jenkins-slave.exe
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To answer my own question, after having contacted the plugin developers:
There is no actual way to download the exe from Jenkins directly, the slave.jar gets it from the master via the remoting protocol. I have created a request to be able to download it via an URL (as suggested by the developer), so it might be available in the future.
Right now the executable is a renamed Windows Service Wrapper binary: https://github.com/kohsuke/winsw so I used this binary the same way.
Regarding the configuration used by WinSW and XML files, I used the one from the GitHub repository https://github.com/jenkinsci/windows-slave-installer-module. The versions are compatible in terms of the configuration.
So basically I download the exe, get the private key from Jenkins and create the service using the configuration from the original plugin. Then I install the service using jenkins-slave.exe install.
Step by step:
java -jar slave.jar -jnlpUrl http://jenkins...slave.jar file from Jenkins (gotten from the JNLP command)http://repo.jenkins-ci.org/public/com/sun/winsw/winsw/2.1.0/winsw-2.1.0-bin.exejenkins-slave.exe install