How to move files from a broken Mac to a new Mac?

Since your 2011 MacBook Pro is not turning on now, data recovery from it is not so easy. You have another 2012 MacBook Pro to use, maybe you can try Target Disk Mode to move files from the broken Mac to the new Mac.

Simply hold down the Power button then immediately hold the T key and see if the dead Mac can boot into Target Disk Mode. If yes, you can transfer files from the dead Mac to a new Mac. If your dead MacBook can't boot into Target Disk Mode, try to boot it into macOS Recovery Mode.

Turn on the MacBook and immediately press down the Command + Option + R keyboard keys together (not regular Command + R). Hold them until a spinning globe shows on the screen. Your Mac is booting into Internet Recovery Mode which enables the macOS Recovery to upgrade to the latest compatible macOS.

If your dead Mac can enter macOS Recovery Mode, then do the following to learn how to recover data from a Mac that won't turn on:

Step 1: In Mac Recovery Mode, choose a network for your Mac and make your Mac connected to the Internet all the time.  

Step 2: Open Terminal from the Utilities drop-down menu. Type the following command in Terminal and press Return. This will launch iBoysoft Data Recovery, a professional data recovery app that has the ability to get files off a Mac that won't boot.sh <(curl http://boot.iboysoft.com/boot.sh)

Step 3: After launching iBoysoft Data Recovery, follow the wizard to scan data from Mac hard drive and recover data.

If you can't boot into macOS Recovery Mode or Target Disk Mode, it's recommended to send your MacBook Pro to a nearby data recovery lab to recover files. Do not plug the old hard drive into a new one by yourself since any mistakes can cause physical damage to your drive and data recovery can be impossible.