How does the President make undeclared war?

It's simply a matter of calling it something else. Military Action can be called for by a president in response to virtually anything.

Its up to congress to approve of the military action if it is going to require long term funding or diplomatic statements of cause.

The only step to calling it war after an approved military action has been sanctioned is an additional official piece of paper declaring that it is a war and presenting it to other countrys as an explanation of why you are engaged in the military action in an effort to convince them not to try and stop you.

Declaring something a war is typically only done if the scale of the engagement is so large that many other nations are threatening to become involved; in which case you want an actual declaration of war to outline for them why you are engaged and trying to convince others to either stay out or help.

If you don't care about other countrys involvement one way or the other, than there is virtually no reason to ever put the label of war on it with a declaration.

Military engagements happen fairly often, declarations of war are far less frequent.