One of my cooks died, how do I get rid of the corpse?

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]"    
        Id.i                 807  
        Id.u                 4708509  
        Type                 DogHandler  
        SubType              1  
        Pos.x                5.89334  
        Pos.y                17.7848  
        Or.x                 -0.810015  
        Or.y                 -0.586410  
        Walls.x              -1.00000  
        Walls.y              1.00000  
        Damage               0.0378701   << Injured
        Vel.x                -0.228300  
        Vel.y                -0.165277 
        ...

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.

You're going to need to build a morgue to put the body. A morgue must be indoors, and contain at least one morgue slab. I'd put a few more than one in there, however, because riots can get pretty nasty.

As of Alpha 11, a hearse will come to your prison, and the dead will be taken away for a proper burial. However, you still need to put the body in the morgue first.

Failing that, the game might have gotten stuck like this. Prison Architect is an alpha still, and one of the big things I have noticed is that sometimes, things like this get stuck. I once had a prisoner awaiting lockdown, sitting in handcuffs for months after his sentence was up. No matter how hard I tried, the guards wouldn't take him away, or even recognize him. If everything is set up correctly, and it still won't work, it's likely you've just encountered a bug.