How to escape HMP-Irongate

You are on the right track. HMP-Irongate is the worst.

You will need a Bed Dummy, Guard Outfit, Grappling Hook, Makeshift Raft, Plastic Staff Key, and cutting items (plastic knives OK, Sturdy Cutters better)

Recipe for Mackshift Raft (requires 80 INT):

Sail           = Timber + Bed Sheet
Raft Base      = Balsa Wood + Balsa Wood + Length of Rope
Makeshift Raft = Raft Base + Sail + Length of Rope

Prep:

1. Label the 4th guard something like "444444", "Staff Key" or "Red" as the key pattern is always the same. It is recommend (but optional) to get the mailman job so the cameras always ignore you. It is also best to get a (Durable) Contraband Pouch to bypass the scanners (doesn't work with console or mobile).
2. Get a Screwdriver, Powered Screwdriver, or a Plastic Knife (slowest), and get rid of the vent above your cell. Block the camera with a Tube of Toothpaste or Shaving Cream. After receiving the Vent Cover, cover the open vent with it to stay safe with a lower percentage. Craft a few Fake Vent Covers with two papier-mâché, and if you are you using the normal vent cover, replace it with the fake one. Make sure that if your last one breaks, load your last save to avoid being placed in Solitary, or use the normal one. It is also best to use your vent system to store your items to stay safe of shakedowns.
3. Craft a Cup of Molten Chocolate, then obtain a Wad of Putty and Molten Plastic. Knock out the guard with the Staff Key with the molten chocolate, and take his Guard Outfit. Use the key to craft a Staff Key Mold with the Wad of Putty -- make sure you put the real key back on him to avoid being placed in solitary.
4. Craft all the necessary items needed for your escape, and go up to the vents to cut the slats going to the right. You must have access to the ladder on the right side.

Night of the Escape:

1. Put the Bed Dummy in the bed, block the camera, and go into the vent. Collect everything you need. Put your Guard Outfit on, and go towards the ladder. You are safe to go to the roof at 23:00 without getting sniped. Go upwards, and when you see a pole, hop on to the ground when no guards are looking.
2. Go to the left and circle around the prison. Go down when you see a red key door on the fence. You should find a building with a ladder and a Prisoner Stash. Go up to the ladder and cross the pole to the left. Cut the fence. Once it breaks, use the Grappling Hook on the wall going down.
3. Go right and down a bit until you find a little wood platform. Use your Makeshift Raft at the edge, and escape!

Source: https://theescapists.fandom.com/wiki/HMP-Irongate