Hi John,
There are just two ways to solve this issue: reactivate your license online on the second machine or copy your license file from the first to the second machine, as Mike has shown.
If PixInsight still asks for a license reactivation, then you haven't copied a valid PixInsightLicense file to the correct folder on your second machine. The reason may be here:
I removed PI and all existing licenses and then deleted the Pleiades folder under user/appdata/roaming
If a Pleiades folder does not exist under Roaming, you have to
create that folder manually, then copy your PixInsightLicense file to it.