If Lightroom is still running on the old computer, then programs such as Belarc Advisor ought to be able to find the serial number on your old computer from wherever it is stored.
If that doesn't work, and you registered your software with Adobe, you can log onto the Adobe web site and recover the number from there.
And if that doesn't work, um ...........
I would also suggest that you develop a much better mechanism for keeping track of your serial numbers for all of your software, no excuses ...
You should be able to use this tutorial to do it:
It will transfer Lightroom and other CS products, along with settings and license info.
Note that you will probably still need to reactivate / validate the license, but it will be there.
Hope this helps!