It's my understanding that both the laptop and desktop treat the HDMI port as output only. If they were physically set up to receive as fast as they can send then this could be made to work with the help of a team of skilled programmers to write the software. Unfortunately, this has not happened yet and so using the HDMI ports to transfer files is not an option.
However, it is possible to use the USB ports for direct transfer. What you would need is a USB A to USB A cable.
If you don't have this, there are still several other options.
You could connect via local network using either wired or wifi connections or a combination of both using a wifi hub.
You could connect via ethernet cable directly, using a special cable. The usual cable will not work.
You could connect directly via wifi by creating an ad hoc connection.
Finally, the easiest solution is to buy an external drive to copy all the files to and then copy them from the external drive to the laptop. Everyone should have an external drive for back-ups anyway. If your hard drive crashes and the backups are on the same hard drive, what good are they?
The problem with the other solutions offered is that they neglect to consider the size of the external drive you will need. It depends. If you can fit all the files on a 4GB thumbdrive then you probably already have a phone with a 4GB storage you can use. If you need 32GB, you can get an SD card this size that will probably fit in your phone. If you need more than that, then you may as well buy a 1TB or larger external drive which you can use for backups. You can get one that connects via USB or via Ethernet/Wifi.
Finally, more information is needed to give you step by step directions. The process will vary depending on the operating system.
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