You need to have Adobe reset their activation count
Chat/Phone: Mon - Fri 5am - 7pm (US Pacific Time) <=== NOTE DAYS AND TIME
Be sure to remain signed in with your Adobe ID before accessing the link below
Serial number and activation support (non-CC)
Adobe allows you to have the same purchased activated serial number on no more than 2 of your computers.
Are you sure that the same purchased serial number is on only on those two computers - the laptop and the Desktop which has been reworked?
Adobe is the only one who can resolve this type of situation requiring removal of purchased activated serial number trapped on a crashed computer. I am not sure what your answer is going to be since the only place (Adobe Chat) that should help you has a "currently closed" sign, and that sign has been there for weeks.
If you go online, you can find an Adobe Customer Service telephone number listed. I do not know if it is functional so that it can help you with your particular issue.
You could try a free ccleaner run through on the removed computer and then the program install to determine if that will help.
In addition, you could follow that with some manual editing of the registry to make sure the cleanup is complete for Premiere Elements.
Please update us on your progress as your schedule permits.
IMPORTANT Add On...At the end of the path in John's link, there is an Adobe Chat which does not have a currently closed sign. Please try that first.
and let us know the outcome.