For DSM 2.0/2.1 users:
Please contact the support desk at firstname.lastname@example.org to log a ticket with your request and the last 5 characters of your unique license key.
The licence will then be reset by one of your support personnel, allowing you to setup the add-on module on a new computer.
For DSM 4.0 users:
Head into File->DesignSpark Options->Licence and click on Deactivate licence. Make sure you are online. This will reset the key for a new activation.
The re-install steps will be the same as in your licence e-mail. Or, you can follow this tutorial from step 4: https://designspark.zendesk.com/hc/en-us/articles/212216845