As with the other multiplayer escapes, you can do this one locally with two controllers and thus 'solo'. Controlling two characters is a pain, so I suggest doing as much prep work for this on your main account, then bringing in your character in local play only for the parts that require a second character and the final escape (basically the stuff either behind a keycard door, or a multiplayer door.)
To prep, you'll need a fake cyan keycard, fake red key, fake red keycard, 2x wire, and a weather vane (either bought from vendors or found in desks). You're going to need to hoof it around the prison getting the pieces for the monster. Bolts can be found in the basement level, either in the warden's office desk or the other desks in the cells down in this area (you'll need a red key, guard's uniform, and contraband pouch—1x foil and 1x duct tape—to access these areas and check everything without raising the alarm.) Now onto the stuff that needs another person/controller.
The leg can be found in the northeast corner of the prison, on the second floor in the multiplayer room. Just loot the desk in here.
The monster arm will require a cyan key card and medical outfits (1x bleach and 1x guard or inmate uniform) to get into the medic's room of the southwest infirmary and then the key card room beyond (the very north room in the long hallway.) Grab the item from the desk.
Once you have the leg, arm, bolts, and lightning rod (combined 1x weathervane and the 2x wire), grab a red keycard and go to the south-southeast corner of the map where there's the lab. There will be a keycard access point just north of the lab on the floor above. Enter this area and then go up the stairs and outside to the very corner of the exterior area and you'll get a prompt to put the lightning rod in place. Then head to the first floor and the co-op door to the lab. Get both players inside, assemble the pieces of the monster as prompted (the shrouded thing on the table in the middle of the room), and you'll get an escape option.
(Note: this worked for me, but I've seen some reports online that they weren't getting the prompt to put the rod in place before they assembled the monster—once you put his pieces in place your character says something like 'now we just need some electricity' or something.) Make sure you have both players by the lightning rod point in the very NW corner of the red door zone to try and get the prompt; otherwise, build the monster, run over there, place the rod, and then get back to the monster and escape.)