Here is what I see for my LR5 order - I was using the Photography Plan $120/year (billed at $10/month) cloud plan by the time LR 6 came out so I don't have a distinct order for it but I assume you can see a LR 6 order if you got it that way: (boxed products, although for me these are from 2012 and give an error)
(download links for downloaded orders, this is where I saw the LR5 order, below) (serial numbers, but no download links, so went back) I am a cloud member so my clicks are different than yours, but there is a way to get to a list of your orders, then you can click on the LR6 order, then click on a link to download the 6.0 version and see the serial number, which means you could download and install LR 6 even if you didn't have the information from the old computer.
Adobe has an easy way to find the download links if you start at and log in:
