The LR5 serial is relatively easy to locate on your PC if you can't get it online. You'll just need to look at with with a text viewer.
On a Mac it's here:
/Users/[user name]/Library/Application Support/Adobe/Lightroom/Lightroom 5.0 Registration
On Windows it's here:
C:\ProgramData\Adobe\Lightroom\Lightroom 5.0 Registration.lrreg
LR6 doesn't use this method unfortunately.
Instead of logging into the LWS, try logging into the adobe page www.adobe.com , select manage account, select plans and products>view products.
That should show both your LR5 and 6 serials.
The adobe server seems to be exceptionally slow at the moment, so maybe that's why you are struggling with LWS.
If the above doesn't work you might have to contact adobe support directly
Thanks Andrew, All sorted now
Good, glad to hear.