If the other computer isn't working and if you don't have a backup there may be only 2 options.
1) If the hard drive will still function on the old computer you could remove the hard drive from the other computer and buy either a hard drive enclosure or a sata to usb adapter and connect it to the usb port on your new computer and copy the photos from the old hard drive. You can do an internet search to find hard drive enclosures or sata to usb adapters.
2) Extract them from the phone using the software I recommended or other 3rd party software. https://www.wideanglesoftware.com/touchcopy/ is the only one I personally know of that will retrieve photos synced to a phone via iTunes but there may be others.