i also suffered from this problem. i may be able to help you.
solution 1: try new cartridges . if it fails try solution 2.
solution 2: see this video. https://www.youtube.com/watch?v=VlKwGVq4MSE. but instead of using the aluminium chips i prefer cleaning the copper contacts with penny brite cleaner .[it worked for me , my printer was sitting idle for a long time and it damaged the contact surfaces.]. it is non toxic available on amazon. see the video you will understand. if cleaning the contacts does not help then you can try aluminium chips. but remember then you have to watch out for empty cartridges.
solution 3: if above solution does not work then buy new printer head form eBay. remember to buy only if accompanied by setup cartridges. this very important.
remember ink system failure message occurs when the cartridge chip says it has ink but the ink does not flow. hp uses a special ink which also transmits current.