1. Try to use the same instruction you used before (The license from the old computer will never work on the new one). Mainly if you are using the University Student license may not work, but if you install the schools license it will run without any issue. If you let me know the product code you used (so I can get the version and edition), I can tell you how to solve it.
2. Send an email to firstname.lastname@example.org and describe the issue, the version of license needed, provide a screenshot with the licensing window on the new notebook and the Host ID (Mac address of the Ethernet).