As there is no “De-Activate” button in the License window, just install eM Client on your new PC you are now going to use and click “Activate” under license. Sounds like the license has already been deactivated in eM Client on your old laptop so nothing to do then on the old pc.
If you have lost your license, go to the following website and enter your email address to send it.
Lost Activation Key | eM Client
Also if you want to transfer the contents to the new computer, (within eM Client) click “Menu / Backup” or “Menu / File / Backup” which will create a.zip backup file. Then copy that .zip backup file to the new pc and put it in the same location. Then restore that via “Menu / File / Restore” and all your data should then come back. See official support link below for transferring to new computer.
How to move my data to a new computer? - Powered by Kayako Help Desk Software (emclient.com)
You can also download the latest versions of eM Client for Pc & Mac via the following version history page to install on your new pc / mac. https://www.emclient.com/release-history