First, how certain are you that the internal drive is OK? Also, what OS have you been using on your old Mac, and also on your new Mac?
Secondly, can the 2011 machine be booted up? If so, I assume the screen is blank. If that's the case, how do you turn that machine off?
Third, assuming the old machine can be booted up, can you provide the exact year of your 13" MacBook Pro? The reason I ask is that if that machine has a Firewire 800 port, or a Thunderbolt port (your other Mac has 2 Firewire 800 ports), you'll be able to connect the 2011 machine to your "newer" one, and boot that older Mac in "Target Disk Mode". What will happen is the 2011's internal drive will appear on the desktop of your other Mac, and thus you can treat it like any external drive.
Next, if the older Mac cannot be booted up, what you can do is remove its internal drive, and install it inside an external enclosure. I suspect the internal drive is 2.5 inches "thick", so you could purchase a nice, slim Orico external enclosure:
https://www.amazon.com/ORICO-Tool-free-External-Enclosure-Supported/dp/B00B0RD2RA
I have 2 of those, in which I have installed Samsung 850 Pro 512 gig SSDs, and they work very well. The interface to your "newer" Mac will be via USB 3.0, but even if the USB ports on your machine are USB 2.0, it will work fine.
Once you get the drive installed, connect the enclosure to one of your USB ports. Boot up your Mac, and that drive should appear on the desktop. But I am unsure whether or not you'll be able to run programs from that external drive.
If instead you actually need to run everything off that hard drive, you could install it inside your newer Mac, and take out its internal drive and install it inside an external enclosure.
