If you want to make this way of connection working for the both of the servers, you have just to execute the steps above for each one.
Even being a project's need, you should be aware that there are some security implications of using passphraseless keys.
When you store your passphraseless key in a file, you would be doing something like writing it in a piece of paper and showing it
for everybody who has access to the machine.
That is, if someone has the access of your machine as a super user, he or she can easily get it and do whatever. Be careful!