Did you register ADE on the old laptop with an Abobe ID?
If so, you just need to register the new laptop with that same Adobe ID.
If not, you may be a bit stuck.
The Adobe DRM infrastructure has some very undesirable features, and you may have hit one.
When you first run ADE it gives you the option to authorize with an AbodeID, or some other option ('Not Yet'?, I forget exactly how they put it).
That second option effectively authorizes with an 'anonymous' ID which has some very undesirable properties.
ADE does give a tiny bit of warning if you make that choice, but not nearly strong enough.
Any DRM books you buy or borrow get associated with a particular ID when you first process the .acsm and download a DRM .epub file.
After that point it is not possible to change the ID with which the book is associated.
You can copy it between devices (up to 6) associated with the ID and all will go fine, but you can't use it on a device associated with a different ID.
You can change the ID associated with a device (remove the old one, ctrl-shift-D for an ADE computer, ctrl-shift-E for a plugged in eReader device, and associate with a new ID),
but you can't change the ID associated with a book.
An implicit/anonymous ID cannot be associated with any other device, and once an anoymous ID is removed (or lost through loss of device), I do not believe that it is possible to recover that same ID.
Therefore it is not possible to regain access ot the book.
Your best bet may be to try a DRM stripper such as epubee (google it). Do not then sell or give away the DRM stripped book to anyone else as that is not legal.
I'm not sure if that will work except on a device which has authorization for the ID associated with the book; in which case you may still be stuck.