Your games are attached to your account, not the PC.

If you can't use the LAN transfer process to install the games to your new laptop, you can always re-download them from Steam to the new laptop.

Download Steam on the other PC.

First install Steam on your new laptop,

Once installed take the steamapps folder from your old PC and merge it with the Steamapps folder on your new laptop, restart Steam and all your games with be there, installed.

A friend at work today told me that if, on my Steam program listed game, it says "CLOUD STATUS, Up to date", that it'll be able to be brought back on the laptop somehow (going over it more tomorrow). Some of my games say that, some say nothing about "Cloud Status" at all. How do I fix this? Should I post a list of my games here, with which ones have this notification and those that don't?