|Apparently the photosmart D5460 has multiple cartridges that are ink tanks with chips that keep track of the ink level and probably the number of times the cartridge has been used , and one large print head on the printer. |
The print head (a.k.a. printhead) is removable. You could try removing that, wiping off it's electrical contacts, and plugging it back in again.
How to remove print head starting page 179 of the User Guide pdf
You probably already have that User Guide, at the very least on the CD that came with it, if not already on the computer - if not you can download it here:
Other than that,
when I search the web with: photosmart D5460 "Ink System Failure"
I get lots of "hits", for this printer model, and other HP printer models.
It appears that at least some of the time you get the message when the print head (a.k.a. printhead) has failed.
You could go here, scroll down, and find the Product Number on the label on the printer,
then go here and click on the same number to find the parts listings,
however, none of the ones I looked at list the print head (a.k.a. printhead).
The HP part number may be obvious on the print head (a.k.a. printhead) itself when you remove it - you could try searching with that on the web.
If you can't find one, if you have a lot of cartridges you would like to be able to use up, you could search the web for a used D5460 that still works, or the model is still available new, or other HP models may use the same cartridges.
If you'd rather get another printer, I recommend Canon printers - you CAN get new or good used print heads (a.k.a. printheads) for those if you need to, and you're a lot less likely to ever need to.
Or if you'd rather get one that has the most economical ink cartridges, get a Kodak model.