You have a morgue and from the picture I cannot see any problem with it, so it must be a bug causing the Cook to be left in the infirmary. One way to fix the problem is to edit the save game file but remember to take a copy first in case editing it causes it not to load properly.
Search for the text "Type Cook" to locate each cook in the file. For some objects the information is all on one line, other times it gets split into multiple lines, so use a regex like
Type[ ]+Cook[ ]*. The reason to match spaces after the word "cook" is that otherwise you will also find all the
Cookers in your prison kitchens.
Check for a line with
Damage followed by a number, I think
1 represents full damage and any value between 0 and 1 is the level of injury.
E.g in my prison I have an injured Dog Handler
BEGIN "[i 807]"
Damage 0.0378701 << Injured
Changing the value to zero, or removing the line should bring the cook back to life!
If you have any problems add the section of the save game file to your question and I'll see if I can help.
Original answer, before the question was updated showing the morgue:
If you don't want to create a whole new room for the morgue you can allocate space in your infirmary for a couple of slabs.
Change the room type of the top part of your Infirmary to Morgue (red rectangle in my picture) and then add one or two slabs (white rectangles). The chef will be moved to one of them and eventually a hearse will arrive to take him away.
Don't worry about the doors being blocked, people will still be able to go through them.
If you start to get a lot of bodies piling up then you can build a separate, bigger, morgue.