I have a laptop with a broken screen & HDMI output. I'd like to plug this in to my working laptop and view the output.
Is there any software that can do this?
No, the HDMI socket on laptops (well, everyone I have seen) is just for Video (and audio) output only.
What you may want to do instead is use Remote Desktop (if supported in your version) or another remote access solution. You can find many here.
If your screen is so damaged that you can not see anything/configure remote desktop, you may want to just link it up to a TV or another computer monitor temporarily to instal/configure.
I have my laptop connected to an external monitor using its hdmi, use it that way every day. Connect a TV or Monitor with a hdmi input, and boot up, you may not see the bios splash screen but when windows loads explorer it should output to the TV/Monitor automatically.