Lexmark offers two types of cartridges for your printer. The 14 + 15 and the 14A + 15A. Only the cartridges with the "A" suffix are refillable. The cartridges without the "A" suffix are part of the Lexmark return program, and they are less expensive than the refillable versions. Links to instructions for refilling the "A" versions here: http://www.refillinstructions.com/Lexmark/L15.htm and here: http://www.refillinstructions.com/Lexmark/L16.htm
The idea with the return program is that the customer is required to return the empty cartridges to Lexmark using a postage paid envelope, so they will not get refilled. The customer may "forget" to return the cartridges, so I think the cartridge is rendered inoperative by burning a fusible link or something similar, when the printer decides the cartridge is empty. There may be a way to circumvent this, see this thread on another forum and note posts #12 and #14 : http://discussions.virtualdr.com/showthread.php?t=232993
I don't own a Lexmark printer so I cannot test this "tape trick". If you try it, please report if it worked or not. I think the results could be valuable to other Lexmark owners.
I don't know what a reset kit is, but a quick Google search on "Lexmark reset kit" suggests it is a kit of maintenance parts for laser printers, not inkjet printers.
To answer your original question more information is needed, as stated by jtoolman. So does the printer reject the cartridges and refuse to print? Or does the printer accept the cartridges and start to print but puts no ink on paper or produces a bad quality printout?
That was just the info needed to answer your original question.
I think the cartridges have been flagged empty by the printer, so they are rejected by the printer and likely useless now. If you order new cartridges go for the "A" versions and you will have a chance to refill successfully, when the new cartridges are empty.
When your ink cartridge is empty, most people send it off for recycling and buy a new replacement. However, replacement ink cartridges dont come cheap; it costs a lot less to refill the cartridge yourself using a specially designed kit. Inkjet refill kits can be found on the internet and in most shops selling printer cartridges. There are two separate kits available, one for filling black cartridges and one for colour. The black refill kit is a lot simpler as you are only dealing with one colour. The colour refill kits are slightly more complex as you have three separate colours Cyan, Magenta and Yellow, these are the 3 primary colours for printing. Most kits are the same, however some have slightly different instructions specific to a certain brand or type of cartridge. If your printer model/brand is listed on the outside of the box it may be easier to use than other brands. Generally, a refill kit can be used to fill any type of cartridge.