OK, after hearing all that it definitely paints a better picture for me of what is going on.
IMHO I feel you've got real problems in your black. (duh) But I mean the problems that likely only a new head will cure. If your employee let the ink run out AND it's been sitting, then you're likely dealing with ink dried inside the channels of the head, not just on the surface of the nozzles which is far easier to fix. Can you get this cleaned out, I don't know. But at this point you likely have nothing to lose by trying.
Now, FWIW I had a similar issue when I bought my used 1604 this summer from someone that didn't take care of it. In fact, when I went to inspect the machine for the first time it had been sitting for over 3 weeks without being turned on!!!! The black lines were absent of ink and when I finally got that figured out both channels of black only had about 10% of the nozzles firing and the other colors were about 50%. After weeks of hard work, head soaks, flushing lines, and eventually manually flushing the head with a syringe by pushing/pulling fluid through the ink manifold -AND- gently swabbing the head surface itself!!! Eventually I did finally get the black back to about 95% of it's nozzles firing and the other colors are all 98% or better. So there may be hope for you...
So the question becomes should you try this?...I feel you can, but it depends on how committed you are. The quick fix to get your machine back up and printing would be to just replace the head - and depending on your business, might be the best scenario. To be completely honest you're pretty much looking at a new head already. If you do it yourself it will take days, more likely weeks to get everything resolved - if it can be. HOWEVER, I want to stress that if you attempt to clean it yourself please don't blame me should anything go wrong. OK?
The 1204 presents unique challenges because there isn't much room to work in there compared to my 1604. So if it were me, shipping your head somewhere to let them do the ultrasonic cleaning may be your best bet with the least hassle. However, if you want to try it yourself go buy an ultrasonic cleaner from Harbor Freight and go that route. My thinking is that the ultrasonic waves will break up the dried ink inside the channels of the head. From that point I would try to gently push/pull cleaning solution through the head by hooking up a syringe to the head manifold using a piece of drain line. I must stress that GENTLY is the key word. You will not see a "stream" of fluid coming from a DX5 head like you would a DX4 with bigger nozzles. If you're doing it right you'll see it slowly pool on the bottom of the head plate and then drip. Again, not sure what kind of tools you have or parts, but when you run solvent printers you should have all kinds of things on hand like extra ink and drain lines, extra dampers and various syringes and cleaning swabs. (I've basically got a small tool box built around servicing both my printers.)
Realize to clean the head you have may take several attempts of ultrasonic cleaning, flushing, etc. After you put it back in the machine and get it back up and running continue to do print checks and measure any success. If you're gaining nozzles, keep going. You may need to pull it out and do another round of ultra-sonic and cleaning, etc. Once you stop seeing improvement that will likely be the point at which your head will be as good as it can be.
So, ultimately the choice is yours. Go the easy, expensive route and just replace the head. Or go the more involved, cheaper route of cleaning it yourself with no guarantee you will fix it. Sadly, those are your choices - but it is what it is....
Good luck to you.