How can I transfer MS Publisher from my dead laptop to my new pc?

There is no simple way to "move" an Office program from one computer to another.

An installed program consists of files placed in a specific set of folders, sometimes the files are specific to the hardware, ie Intel vs AMD cpu. You can copy the files and folder structure, but you can't deal with hardware specific issues.

During the installation the program may make changes to the windows configuration, things like the "path". (granted, that specific setting is not as important as it used to be).

Finally, MS insists that the program be registered, activated, on the computer to prove to them that you have a paid for license to use the program. This is the stumbling point for you since you say you don't have the key. Office 2013 was still activated to specific hardware. Even when your license allows transfer to new hardware you still have ot activate it on the new machine by submitting the product key. MS expected that we would be respponsible and keep track of your own product keys. Office 2013 did offer you the option of "registering" the key with your email account, but most people did not take advantage of it. If you did, you are one of the lucky few 2013 users who can log in to the MyAccount site and install it on the new machine from inside MyAccount.

Before ms "improved" 2013 we could use a tool to extract the product key. Starting with 2013 the tools only extract the last 5 digits of the code. Enough to match it to your record of the key (kept elsewhere), but not enough to use for activation on any machine.