Q: Using Migration Assistant with no display

I have had my Mid-2010 (2.4 GHz, 4 GB RAM, OS X El Capitan) MacBook for 2 years.

It split in half somewhere in those 2 years.

The MacBook was still usable, but it had to be on a desk. The display had completely separated from the base. It still worked.

I woke up a couple days ago, to find that someone had cut the display cable holding the MacBook together and powering the display.

I bought an Early-2011 13" MacBook Pro to replace the dead MacBook. (Yes, I buy old Macs because the new ones are so expensive.)

The old Mac was backed up semi-regularly, so I have an old backup to use for the new Mac. The problem is that right before the old Mac died, I downloaded a bunch of stuff that wasn't backed up. I really want those files that weren't backed up to be on my new MBP.

Can I use Migration Assistant with an Ethernet cable and transfer data from that, or does Migration Assistant have to be running on the old Mac as well?

I tried to sign in (I'm pretty sure the computer still works, it just doesn't have a display) on the old Mac, plug in the backup drive and hopefully do an hourly backup, but Time Machine didn't do that.

Any suggestions would be GREATLY appreciated.