Broken door knob. no access to hinges, screws. Rental property!

Might have to cut the screw to remove it from the door. You didnt say if the knob was plastic or metal. If its metal and you cut the screw head off. Take the knob outside and heat with a torch to warm it up. It may losen and twist out with 2 pairs of pliers.

Depends on type of hinge, If it swings back open after you closed it then either the cupboard is not level (old style hinge)...
Or the spring inside hinge is bad/broken and not holding it closed. If the door opens back just a bit (1/2 inch or so) see if hinge has adjustable screws.

well if you havent been able to get the cat out by now its dead but looking at the pics only way you can get in is by jigsawing the latch out

I just fixed the problem with my Krups FBC-212. The door would not stay closed because the spring that keeps it in the closed position was broken. I ordered another from one of the small appliances parts sites (about $7 plus shipping) and that fixed it. I will describe how to do it and how to avoid the mistake that made it much harder and longer than it should have.
Remove the black screws at the back side, the ones holding the external black metal case to the silver chassis. There are 9 of these on my oven.
Also remove the 3 screws holding the left-side foot on.
Now you can probably slide the metal case (previously held on by the 9 screws) out from under the black metal front frame on the left side, thereby exposing the broken spring.
If you cannot get it out from under the lip of the front frame, you can take off the right-hand foot, and if that is not enough, you can remove THREE of the 4 black screws just inside the door holding the front frame to the chassis. Do NOT remove the lower left one next to where the flat door hinge piece goes into oven (which was my mistake) because that screw holds a critical black plastic piece in place.
By now you will be able to slide the black case that covers the sides and top backward and outward enough to expose the broken spring. It will be obvious how to attach the new one.

Do you have access to the hinge pins? If yes, remove them - remove the door - replace the handset.

If not, remove the handle by hack saw if necessary ... open it manually ... use a screw driver ... replace the handset.

The door may not be closing all the way letting outside air in causing the ice buildup.
There is a solenoid that operates the door that could have stopped working. Can you hear it?
The hinge on the door could be broken or the bracket it goes in could be broken. You'll have to look to see what's wrong.

Remove the dispenser cover by gently prying starting at the bottom going round and round a little at a time to prevent from breaking it. When it's off the screws will be exposed to access the other components.

The door hinge is probaly broken or could just be the spring. Open the door to the 45 degree position (halfway) and pull on the door. It should come off the hinge assembly. Slide out the oven a little to access the screws for the hinge to see what is broken or wore out.

I agree with your Hacksaw method. Very easy and very fast. You are going to want to replace the knob anyway so this does not happen again. Just Hacksaw that knob as close to the door as you can without damaging the door. From there you are on your way to access to the parts of the lock. Now you can replace that knob and never have this trouble again!!!! One less headache in Life......Hope this helps you ........Joe

I used a hatchet. You have to replace it anyway.