Another company here:
https://www.krollontrack.co.uk/services/data-recovery/mobile-phone-recovery/
Am no expect on data recovery for flash type media on mobiles but from the googling articles it looked almost impossible to recover pictures/files from phones, mobile phones don't even show up on as a accessible drive letter on the pc by design. This is the first issue so data recovery software can't work like normal hdd/ssds do.
I am curious how forensics, kroll or other data recovery companies get around finger print/password encryption, is that not the whole point of encryption otherwise why have encryption in first place?
Anyhow to me just like others suggested maybe if the charge port worked partly at least its possible to charge it? try it see if it gets more fuller or try and borrow a wireless charger but maybe you could replace the actual broken screen ? Quick check on ebay and replacement screens cost around £110.
Vids showing you how to do it in 10 mins:
https://www.youtube.com/watch?v=AyECCSa5hGU
It maybe worth trying to fix the fone yourself rather then spending £500-1k on the recovery, I can't say for sure mind you.
Another potential fix is asking a few mobile fone shops if they can do a recovery by repairing the phone screen just so if can be in a usable state for few minutes, maybe these mobile repair shops got an s6 phone or can temporary use one just so you can get your stuff off it and offer a cheaper and quicker alternative, these mobile phone repairers fix mobiles in minutes its not a sweat for them.
I just googled if you could output the display to the TV via some cable, not sure but someone had the same issue as you and took to a repair shop and got it repaired and data taken off. Scroll to bottom almost its in the comments section.
https://www.mobilefun.co.uk/blog/2015/03/how-to-connect-your-galaxy-s6-s6-edge-to-your-tv/
good luck